CHAPTER 1 : BASIC SKILLS

Transcription

1 Mathematics with Symbols Dr Roger Ni (Queen Mary, University of London) -. CHAPTER : BASIC SKILLS Physical quantities are often represented by symbols - these may be the normal Latin characters ( a, b, c,..., A, B, C,... etc.) or Greek characters ( α, β, γ,... etc.). This greatly simplifies the writing out and manipulations of equations - for eample Becomes Momentum Mass Velocity p m v or more simply still p m v where: p - m - v - represents momentum represents mass represents velocity The above eample illustrates how products of physical quantities may be written in a symbolic form in several different ways. More generally, products may be written as a b or ab or a b or a b whilst quotients (division of quantities) may be written in any of the following ways: a/b or a b or a b or a b. Equations may often contain a miture of Latin and Greek symbols. e.g. c νλ. Symbols may be lower or upper case and may have different meanings - for eample: - p is used to represent momentum whereas upper case P is generally used for pressure. - t is used to represent time whereas upper case T is used for temperature. 3. Symbols may be used to represent both constants (physical quantities whose values are fied, e.g. the speed of light, c, or the ideal gas constant, R ) and variables (physical quantities whose values may vary, e.g. the wavelength, λ, of radiation). 4. When typed, Latin characters used to represent quantities should be italicised - however, - the Latin v is sometimes not italicised, as it begins to look like the Greek nu, ν. - symbols may not be italicised if the typist is careless / lazy / short of time!

2 Dr Roger Ni (Queen Mary, University of London) -. Order of Evaluation of Epressions In the absence of brackets, the order of evaluation is: powers before multiplication/division before addition/subtraction. These rules for the order of evaluating epressions also apply when using symbolic mathematics e.g. a b c a bc the multiplication of b and c is carried out before the addition to a To avoid any ambiguity brackets (parentheses) should be used, since the epressions inside the brackets are always evaluated first (i.e. the brackets override the normal order of priority described above). For eample, to indicate in the above epression that a should be added to b before the multiplication by c it should be re-written as follows: (a b) c or (a b)c They may also be used, for eample, to clearly differentiate (a b)/c from a (b/c) d [a (b/c)] from d a (b/c) Commutative & Associative Laws When several quantities are multiplied together, the multiplications may be carried out in any order. i.e. multiplication obeys both the commutative and associative laws: Commutative Law : y y Associative Law: ( y) z (y z)

3 Dr Roger Ni (Queen Mary, University of London) -.3 Brackets Multiplication The notation A( y) is shorthand for A Ay, i.e. A( y) A Ay A ya ( y)a This concept may be etended to any number of terms added or subtracted within the bracket, e.g. A( y z) A Ay Az. Note also the following application of this rule: (y) (y y) y y y.. which confirms that multiplication is both commutative and associative, since (y) ( y) ( ) y - assuming multiplication is associative ( ) y - assuming multiplication is commutative y... which has already been demonstrated to be true. For two brackets multiplied together, repeated application of the rule gives: or, equally (A B) ( y) A( y) B( y) A Ay B By (A B) ( y) (A B) (A B) y A B Ay By Note especially the manipulation of signs in the net two eamples: (A B) ( y) A( y) B( y) A Ay B By (A B) ( y) A( y) B( y) A Ay B B( y) A Ay B By The complete removal of brackets from an epression is known as epansion ; the opposite process whereby a common factor is taken out and brackets are introduced is known as factorisation. Factorisation can always be carried out when every term has a common factor, e.g.

4 Dr Roger Ni (Queen Mary, University of London) -.4 A Ay Az A ( y z) but can also be carried out in other cases, e.g. (A Ay 3z) A ( y 3z ) A One special type of factorisation that you will come across is that associated with certain types of quadratic function, e.g. a a ( a)( a) ( a) a ( a)( a) Division Division of brackets follows the same rules because division is equivalent to multiplication by the reciprocal of the divisor, e.g. ( y z) 5 5 ( y z) y z esting of Brackets When epanding "brackets within brackets" (nested brackets) then the innermost brackets should be removed by epansion first of all. When writing epressions involving nested brackets, different types of brackets are often used to help clarify which brackets together form a pair. For eample, [ y (3 ) ] [ y 6 ) ] y 4 5 y

5 Dr Roger Ni (Queen Mary, University of London) -.5 Fractions A fraction is a ratio of two quantities - the numerator on the top line and the denominator on the bottom line. A fraction remains unaltered in value if the top and bottom lines are both multiplied by the same quantity. i.e. bn an n b n a b a Note - that the entire contents of the top and bottom lines must be multiplied by the quantity concerned. i.e. cn bn an n c b an n c b n a c b a ) ( ) ( The basic rules for combining fractions are summarized below : Multiplication : bd ac d c b a Division : bc ad c d b a d c b a Addition / Subtraction : for two factions to be combined by addition or subtraction they must possess the same denominator. If so, then ( ) c b a c b c a ± ± If the fractions do not initially have a common denominator, then they must first be manipulated (by multiplication of top and bottom lines by the same factor) so that they do have identical bottom lines - the best choice for the common denominator is usually the lowest common multiple of the denominators of the individual fractions. ( ) bd bc ad db cb bd ad d c b a or ( ) 3

6 Manipulation of Algebraic Equations Dr Roger Ni (Queen Mary, University of London) -.6 The key things to remember about an equation are that: an equation is a statement that two things are equal. an equation remains correct as long as you perform the same operation to both sides. For eample, if y then a y a and a y a a ay and a y a a y a (both sides raised to the power of a ) y (a special case of the preceding, with a ) Caution As noted in the preceding section, if y then but y If z then z y (a far too common error!) y Instead, if you need to take the reciprocal, the fractions on the right-hand side must first be combined, which (as noted just above) requires them to have a common denominator, i.e. y ( y) y y y y hence z y ( y) z y y z ( y)

7 Dr Roger Ni (Queen Mary, University of London) -.7 Eample Consider the equation : d M u v To make v the subject of the equation : (i) multiply both sides by v v d M u v v v d M u (ii) divide both sides by d v d d M u d v M u d Eample Consider the equation : v u at To make a the subject of the equation : (i) subtract u from both sides v u ( u at) u v u at (ii) divide both sides by t v u t at t v u a t

8 Miscellaneous Mathematical otation Common Mathematical Symbols Dr Roger Ni (Queen Mary, University of London) -.8 For a table of common mathematical symbols (,, etc.) see the glossary located inside the backcover of this manual. Subscripts Subscripted numbers or letters have no mathematical meaning (whereas superscripts do have)... they are simply labels. For eample, if a molecule may occupy a series of energy levels of increasing energy then the energies of these levels may be denoted ε, ε, ε 3, etc. In this eample, ε i denotes the energy of the i-th energy level. Factorials By definition: n! n (n ) (n )... ( a total of n terms) Eample : 3! 3 6 Special cases : 0! Summations When a quantity or function is defined by a sum of many similar terms it is often much more convenient to write it in an abbreviated fashion: this may be done using the summation symbol, Σ, and by defining a general term. 3 e.g. Consider F()... ( e - see page.8 )! 3! This may be abbreviated to F() n n 0 n! where the numbers above and below the summation symbol represent the limiting values of the variable, n (i.e. n 0,,, 3, 4,... - in this case the series is infinite). In this eample, n n! is the general term of the summation.

9 Dr Roger Ni (Queen Mary, University of London) -.9 Products When a quantity or function is defined by a product of many similar terms it is sometimes written in an abbreviated fashion: this may be done using the product symbol, Π, and by defining a general term. e.g. Equilibrium constant, K p ν Pi P o where P i is the partial pressure of component i and ν i is the coefficient in the stoichiometric chemical equation. i Limits of Functions A limit of a function is the value towards which it is tending, as the variable upon which the function depends approaches some specified value. e.g. What happens to the value of the function f() as becomes very large?... the answer is that as "tends to" infinity, then f() tends to an infinitely small value, i.e. the "limit" (or limiting value) of the function as is zero. This may be abbreviated in either of the following ways : as, f() 0 lim ( ) 0

10 Dr Roger Ni (Queen Mary, University of London) -.0 Indices (Powers) Definition n... ( n terms to be multiplied out; n is called the inde or eponent) e.g ; ; Multiplication In general : A. B AB Proof by eample :. 3 (. ). (..) Numerical eample : Division In general : A B A B Proof by eample : 3... ( ) Numerical eample : 3 3 / Addition and Subtraction Epressions such as A B or A B cannot in general be simplified further. However, such epressions can often be factorized, e.g. A B A B ( ) A A ( (B A) ) Power of Zero It is easy to prove that any quantity raised to the power zero is one: A / A (since any number divided by itself gives unity) but A / A (A A) 0 0

11 Dr Roger Ni (Queen Mary, University of London) -. egative Indices In general : A A ( and A A ) Proof : A (0 A) 0 A A Numerical eample : 3 /3 /9 /3 /(/9) 9 3 Powers of Powers In general : ( A ) B AB Proof by eample : ( 3 ) 3. 3 (..).(..)... 6 Numerical eample : (3 ) Powers of Products In general : ( y ) A A y A Proof by eample : (y) 3 (y). (y). (y).y..y..y (..).(y.y.y) 3. y 3 Numerical eample : ( 3) and ( 3) Fractional Indices A number raised to a fractional power is called a root. Thus: / the "square root" /3 3 the "cube root" The operation of taking the root is the inverse of taking a number to the corresponding power, i.e. it undoes the effect of taking the power. The opposite is also true. So, for eample, and 4 / 4 ; since 4 8 /3 3 8 ; since 3 8

12 Dr Roger Ni (Queen Mary, University of London) -. ( 3 ) /3 3 ( 3 ) and ( /3 ) 3 ( 3 ) 3 A Proof : ( A ) /A A Other numerical eamples 4 3/ (4 3 ) / 64 / 64 8 or 4 3/ (4 / ) 3 ( 4) It follows that the re-arrangement of equations involving powers often involves taking fractional powers of both sides: e.g. if a 3 M u d then to make a the subject of the equation, we raise each side to the power / 3 a (a 3 ) /3 M u d / 3 Summary of Rules for Manipulating Indices Note - even though the definition of n given at the beginning of this section presumes that n is an integer these rules for manipulating indices apply for all possible values (integer and non-integer) of the indices. A. B AB A A A B A A A B ( A ) B AB (y) A A y A

13 Dr Roger Ni (Queen Mary, University of London) -.3 Indeation of Brackets (y) n y. y. y.... n times ( n times) (y.y.y.... n times) n. y n ( y) ( y) ( y) ( y) y( y) y y yy y y ( y) 3 ( y) ( y) ( y) ( y y ) ( y y ) y( y y ) 3 y y y y y y 3y y 3 ( y) 4 ( y) ( y) 3 ( y) ( 3 3 y 3y y 3 ) ( 3 3 y 3y y 3 ) y( 3 3 y 3y y 3 ) y 3 y y 3 3 y 3 y 3y 3 y y 6 y 4y 3 y 4 Note : the pattern of the coefficients that is beginning to emerge - this enables us to predict the coefficients for higher order epansions using Pascal's triangle. ( y) ( y) ( y) ( y) ( y) ( y) Each line of the triangle is obtained from the previous line by first imagining there are zeros on either end of this previous line, and then summing up the numbers in pairs - in each case the sum is written on the line below, half way between the two numbers of the pair from which it was derived.

14 Dr Roger Ni (Queen Mary, University of London) -.4 Scientific otation - Representation of Large and Small umbers in Terms of Powers of Ten In scientific notation a number is epressed as the product of two numbers: y 0 n. The first term, y, is usually a number between and 0 ; the second term, 0 n, is some integer power of 0. Thus: Similarly, a number less than can be written as a number between and 0 divided by a power of 0, which is equivalent to multiplication by a negative power of 0. Thus: Why do we do this? - it is simply a matter of convenience when working with very large and very small numbers. For eample, is easier to write than is easier to write than Other eamples: / / / For numbers greater than the value of the inde n is the number of places by which the decimal point is moved to the left to obtain the number in scientific notation: For numbers less than the negative value of the inde n is the number of places by which the decimal point is moved to the right to obtain the number in scientific notation:

15 Dr Roger Ni (Queen Mary, University of London) -.5 To convert a number in scientific notation to the usual form the procedures are inverted: ote: 0 n is usually abbreviated to 0 n, e.g Eamples of Calculations using Scientific otation In the following eamples, the final answers are adjusted to a precision appropriate to the type of calculation and the number of significant figures specified for the numbers involved - this topic is dealt with in the Essential Skills for Chemists course. Addition and Subtraction When adding or subtracting two numbers they must first be converted to the same power of ten: [ ] [ ] [ ] [ ] ( ) (to 4 s.f) [.34 0 ] [ ] [ ] [ ] ( ) (to 4 s.f) Multiplication The first terms are multiplied together and the powers of ten are added together (cf. A. B AB ): [ ] [ ] ( ) (0 3 0 ) [ ] [.34 0 ] ( ) (0 3 0 )

16 Dr Roger Ni (Queen Mary, University of London) -.6 Division The first terms are divided and the powers of ten are subtracted (cf. A / B A B ): Powers The first terms are raised to the power and the powers of ten are multiplied (cf. ( A ) B AB ): [ ] [.3] 0 (3 ) [ ] 3 [5.67] 3 0 ( 4 3)

17 Significant Figures and Decimal Places Dr Roger Ni (Queen Mary, University of London) -.7 It is sometimes necessary (or desirable) to specify a number to only a certain precision : there are two common methods to do this. Numbers are said to "be rounded to" or "given to" a certain number of significant figures (s.f.) or decimal places (d.p.). Fied Decimal Place Format This format may only be used when numbers are written using normal decimal notation - it cannot be applied to numbers written in scientific notation. The "number of decimal places (d.p.)" corresponds to the number of digits after (to the right of) the decimal point. The value of the last digit given depends upon the value of the following digit prior to the rounding. If the following digit : (i) falls in the range 0-4 then the last digit is left unaltered (ii) falls in the range 5-9 then the last digit is increased by one (i.e. rounded up) Eamples: Unrounded To 0 d.p. To d.p. To 4 d.p Fied Significant Figure Format This format may be used when numbers are written either using normal decimal notation or in scientific notation - it is a more generally useful format. The "number of significant figures " (sig.fig, or s.f.) corresponds to the number of significant digits given in the non-eponent part of the number (ecluding any leading zero-digits). The value of the last digit given depends upon the value of the following digit prior to the rounding. If the following digit : (i) falls in the range 0-4 then the last digit is left unaltered (ii) falls in the range 5-9 then the last digit is increased by one (i.e. rounded up) Eamples: Unrounded To s.f. To 4 s.f

18 Dr Roger Ni (Queen Mary, University of London) -.8 The Eponential Function The eponential function is an etremely important function in science and nature; it appears in many scientific equation including, for eample, in the Boltzmann equation (which defines population distributions of particles amongst energy levels) and in various integrated rate equations in chemical kinetics. Definition : the eponential function is denoted by e and is defined by the infinite series : Key points: 3 e... i.e. e! 3! n it may also be written as ep or ep() n n 0!... but not E this function is also an eample of a power: i.e. e is " e taken to the power of " e is a real number, approimately equal to.7 (see below) The value of the real number e may be obtained by setting in the definition. e e!... 3! (to 8 s.f.) Combinations of eponentials may therefore be manipulated in eactly the same fashion as any other power, i.e. using the rules summarized on page.. e.g. e. e e ( ) e 3 (e ) 3 e 6 The graphical form of the eponential function is illustrated below: f() e ep() : for > Limits : f(0) f() as ote : other functions of the type f() n (with n >) will have a similar form. f ( ) e However, in chemistry it generally occurs in one of the forms illustrated overleaf.

19 Dr Roger Ni (Queen Mary, University of London) -.9 The standard eponential decay function: f() ep( ) : for > 0. Limits : f(0) f() 0 as f ( ) Eample : Change in reactant concentration with time for a first order reaction The negative eponential function for the reciprocal of a variable: f() ep( /) : for > 0 Limits : f() 0 as 0 f() as Eample : Change in population of an energy level with temperature. f ( )

20 Dr Roger Ni (Queen Mary, University of London) -.0 Logarithms Definition : If n y (where n is some pure number) then y log n General characteristics and y is said to be the logarithm of to the base n. The operations log n and taking " n to the power of..." cancel each other out if applied successively - the functions are therefore a pair of inverse functions. i.e. log n (n y log ) y and n n Important : you can only take the log of a dimensionless quantity (a quantity without units) - the result is also dimensionless. Mathematical operations with logarithms It is sometimes necessary to rearrange epressions involving logarithms without finding their actual values using a calculator. The rules to use apply to all types of logarithms and they are: log( y) log( / y) log( b ) log() log(y) log() log(y) b. log() from which it also follows that... log() 0 log( / y) log(y) These rules may be used in combination to simplify epressions involving multiple log terms - for eample: c log(c) d log(d) a log(a) b log(b) c C D log a A B d b

21 Dr Roger Ni (Queen Mary, University of London) -. In principle n can be any number but only two types of logarithms are in common usage:. Logarithms to the base 0 [written lg or log or log 0 ], where n 0. so from the general definition : if 0 y then y log 0 whilst log 0 ( 0 y ) y and 0 log. "Natural logarithms [written ln], which are logarithms to the base of the number known as e ( e , the eponential"). so from the general definition : if e y then y ln whilst ln ( e y ) y and e ln Given a logarithm, it is sometimes necessary to find the number which has this logarithm (sometimes called the antilogarithm or antilog). To do this we need to apply the inverse function, i.e. we simply need to take the power of the base number. For instance: If ln.34, then what is? e ln e log 0 7.8, then what is? 0 log Similarly, equations involving logs can be re-arranged by application of the inverse power function e.g. o G RT ln K o G ln K RT K e G RT o whilst equations involving eponents can be re-arranged by application of the inverse log function. e.g. k Ae E a RT E a e RT k A E a k ln RT A k E a RT ln A

22 Dr Roger Ni (Queen Mary, University of London) -. Special characteristics of the log 0 and ln functions. As noted above if 0 y then y log 0 This property is useful for relating the values of logs with the eponents of numbers written using scientific notation. For eample suppose we have a number where 0 3 < < 0 4, then 3 < log 0 < 4. e.g. log 0 ( ) log 0 (6.0) log 0 (0 3 ) Natural logarithms have special properties which will emerge from calculus. The values of natural logs are also related to the log 0 values. The relationship between them is: ln() ln(0) log 0 ().303 log 0 () Logarithms and hydrogen ion concentrations As noted above, the logarithm to the base 0 is a convenient function for representing quantities which vary over many orders of magnitude. A good chemical eample is the representation of hydrogen ion concentration by means of the ph scale. In aqueous solution the hydrogen ion is best represented as H 3 O (as it eists almost eclusively in this form) and, making certain approimations and using the chemists normal units for concentration, then the ph value is given by: ph log 0 [H 3O ] mol dm -3 Note - [H 3 O ] is a concentration and has units of mol dm 3 - the units will therefore cancel on the top and bottom lines to give a pure number and the log of this pure number may be taken. The ph value obtained is also dimensionless. In aqueous solution [H 3 O ] values typically range from.0 mol dm 3 (in.0 mol dm 3 strong acid, e.g. HCl) to mol dm 3 (in.0 mol dm 3 strong alkali, e.g. NaOH). The corresponding ph values range from 0 (strong acid) to 4 (strong alkali). Other log scales (p-scales) More generally, the "p" is used in various abbreviations to represent an operator which corresponds to calculating the negative log of any quantity that follows the p-symbol eamples include ph, poh, pk a, pk b, pk w. For eample, pk a, log 0 K a.

23 Dr Roger Ni (Queen Mary, University of London) -.3 Solving Equations A single equation can generally be solved if the values of all variables ecept one are known - a solution is obtained by finding the value for this last variable which satisfies the equation. Quadratic Equations Quadratic equations are equations of a single variable (say ) which may be arranged into a form which only involves positive powers of up to order, i.e. they are equations of the general form : a b c 0, where a, b and c are constants. There are two standard methods for solving such equations :. Some (but not all) quadratic equations may be readily factorized upon inspection to give a product of two linear epressions in e.g. 3 0 ( )( ) 0 ( ) 0 or ( ) 0 or. All quadratics may be solved using the general solution which for a quadratic of the form : a b c 0 is given by: b ± b 4ac a Note - a polynomial equation of order n always has n solutions (e.g. a quadratic, with n, has two solutions). Equations may need some rearrangement before they are clearly revealed as quadratic equations which can then be solved by one or other of the two methods described above. e.g. ( ) e.g. w 4 3 w ( where w )

24 Dr Roger Ni (Queen Mary, University of London) -.4 Simple Simultaneous Equations A set of n independent simultaneous equations may be solved to obtain the values of n variables. The simplest case is when n i.e. two equations and two unknown variables. e.g. 3 y 7 3 y 5 To obtain the values of and y Step : multiply the equations by constants so as to obtain the same coefficient for one of the two variables. 3 y 7 multiply both sides by y 3 y 5 multiply both sides by 6 y 0 Step : subtract one equation from the other; the variable which has the same coefficient in both equations will cancel out. 6 9 y 6 y 0 (6 6) (9y ( y)) 0 y Step 3 : solve for the remaining variable, then substitute this value back into either of the original equations to get the value of the other variable. y y (3 ) 7 4 i.e. the solution is and y

25 Dr Roger Ni (Queen Mary, University of London) -.5 Other Equations The best approach in general is to first rearrange the equation to make the unknown the subject of the equation; then and only then substitute in the values for the other variables and constants to obtain the value of the unknown (and its units!). Eample : Calculate the velocity of an electron which has a kinetic energy of J. [ m e kg ] The relevant equation is : E k ½ m v Step : make the velocity, v, the subject of the equation. E k ½ m v E v k m E v k m Step : substitute in the known values for m and E k. v J kg J kg kg m s kg - v m s Eample : Calculate the hydrogen ion concentration in a solution with ph 5. The relevant equation is : ph log 0 [H 3O ] mol dm -3 Step : make [H 3 O ] the subject of the equation. [H 3O ] ph log 0-3 mol dm ph 0 ph [H 3O ] log 0 mol dm [H 3O ] -3 mol dm -3 [H ph -3 3O ] 0 mol dm Step : substitute in the known value for ph. [H ph O ] 0 mol dm 0 mol dm [H ph O ] 0 mol dm mol dm

26 Dr Roger Ni (Queen Mary, University of London) -.6 Partial Fractions If the bottom line (denominator) of a fraction is a polynomial function (e.g. it contains powers of ) and this polynomial may be factorized, then the fraction can be transformed into a sum of simpler fractions - the so-called partial fractions. The procedure for doing this is illustrated by the following eample: e.g. 5 6 Now 6 ( )( 3) so it should be possible to write the fraction as the sum of two simpler fractions, where the denominators of these partial fractions are the factors of the polynomial. i.e. 5 A B 6 ( ) ( 3) where A and B are constants whose values we need to find. We can do this, by combining the two partial fractions using the normal procedures for addition of fractions: A B ( ) ( 3) A( 3) B( ) ( )( 3) ( 3)( ) A( 3) B( ) ( 3)( ) A( 3) B( ) 6 and we know that this must be equal to 5 6, whatever the value of might be. So, A ( 3) B( ) 5 for all values of. Consider (i), then 5 A 0 B 5 A (ii) 3, then 0 A 5 B 5 B i.e. 5 6 ( ) ( 3) ( ) ( 3) One application of partial fractions is in simplifying the integration of functions (see section 5): e.g. 5 d d d 6 ( ) ( 3)

27 Dr Roger Ni (Queen Mary, University of London) -.7 Proportionality Proportionality is concerned with epressing a simple relationship between two physical quantities (i.e. between the two variables representing such quantities in a mathematical equation) when all other variables which might otherwise affect their values are kept constant. The relationship may be a linear one, i.e. y constant y is then said to be (directly) proportional to, a relationship which is written : y The relationship may be a reciprocal one, i.e. y constant / y is then said to be inversely proportional to, a relationship which is written : y / Eample : The ideal gas equation ( PV n RT ) may be re-arranged to epress how the gas pressure, P, is related to the other gas variables (volume and temperature) when the amount of gas involved is fied (i.e. n is constant). To do this: we first make P the subject of the equation. nrt P V nr if the volume, V, is kept constant P T constant T V i.e. P T (with constant n and V ) e.g. if you double T then P will also double. if the temperature, T, is kept constant i.e. P /V (with constant n and T ) P ( nrt ) constant V V e.g. if you double V then P will halve in value. Eample : The equation for the vibrational frequency of a molecule, ν, ehibits a more comple type of proportionality relationship. The equation for the vibrational frequency is: ν π k µ It follows that if k is constant, then ν constant µ or ν µ i.e. ν is inversely proportional to the square root of µ.

28 Dr Roger Ni (Queen Mary, University of London) -.8 This second proportionality relationship may be used to calculate the effect of isotopic substitution on vibrational frequencies. Two isotopic variants of the same molecule, with reduced masses µ and µ, will have the same value of k (force constant). Their vibrational frequencies (ν and ν ) are therefore related by: ν µ ν ν µ µ Where does this come from? The proportionality relationship indicates that: i.e. ν µ constant ν constant µ So in the comparison of two molecular states when k is constant: ν µ ν µ and rearrangement gives: ν ν µ µ So, for eample, if µ µ ν ν µ µ ν ν ( 0. ν ) ν 707

29 Dr Roger Ni (Queen Mary, University of London) -.9 Equations of Classical Physics The following is a list of some of the key equations of classical physics which are also routinely used in chemistry. Classical Mechanics Velocity change in distance / time d v dt Acceleration change in velocity / time a dv dt Momentum mass velocity p m v Force mass acceleration F m a Work (Energy) force distance moved W F pressure change in volume W P V Kinetic Energy ½ mass velocity E k ½ m v ½ momentum / velocity p E k m Pressure force per unit area force / area P F A Electromagnetic Radiation Velocity wavelength frequency c λ ν Properties of Matter Density mass / volume m ρ V Heat energy heat capacity change in temperature q C T Ideal Gas Equation: Pressure volume no. of moles ideal gas constant temperature P V n R T or Pressure volume no. of molecules Boltzmann constant temperature P V k T

30 Dr Roger Ni (Queen Mary, University of London) -.30 SI Units (Système International D Unitès) Base Units The system depends upon certain base units. These form a set of independent units in terms of which all physical quantities can be epressed in a consistent manner. The base units most frequently used in chemistry are the following. Physical Quantity ame of SI Unit Symbol for SI Unit Length metre m Mass kilogram kg Time second s Electric current ampere A Temperature kelvin K Amount of substance mole mol Note - by convention, the abbreviations for units named after people (e.g. Ampere, Kelvin) begin with a capital letter. Derived Units Each physical quantity is denoted by only one SI unit, which is either the appropriate base unit, or the appropriate derived unit formed by multiplication or division of two or more base units, some eamples of commonly-used derived units are given below, together with their definitions in terms of the base units, i.e. their dimensions Physical Quantity ame of SI unit Symbol for SI unit Definition in Base Units (alternative formulation) Force newton N kg m s Energy (work) joule J kg m s Electric charge coulomb C A s Electric potential volt V kg m s 3 A Frequency hertz Hz s Pressure pascal Pa kg m s ( N m ) Power watt W kg m s 3 ( J s ) Note : (i) different units are separated by spaces, e.g. N m not Nm ; J s not Js (ii) we often enclose a quantity in square brackets [ ] if we wish to refer to its units, e.g. What are the units of kt? [kt] J K K J (i.e. the units of energy).

31 Dr Roger Ni (Queen Mary, University of London) -.3 on-si Units The most common non-si units which you are likely to meet are the Ångstrom ( Å ) and the calorie ( cal ). (i) an Ångstrom is a unit of length, which is very convenient for bond distances. Å 0 0 m 00 0 m 00 pm m 0. nm (ii) a calorie is a unit of energy : it is inconvenient, but it is still quite widely used in the US. cal 4.84 J How to Work Out Dimensions The dimensions of any quantity can be worked out from the physical law which defines it. For eample, [ Velocity ] [ Distance ] / [ Time ] m / s m s [ Acceleration ] [ Velocity ] / [ Time ] m s / s m s [ Force ] [ Mass ] [ Acceleration ] kg m s kg m s [ Energy ] [ Work ] [ Force ] [ Distance ] kg m s m kg m s [ Pressure ] [ Force ] / [ Area ] kg m s / m kg m s. Prefies To avoid numerical values being inconveniently small or large, fractions or multiples of SI units may be constructed using the following prefies. Factor Prefi femto pico nano micro milli centi deci kilo mega giga Symbol f p n µ m c d k M G Note : ) Most of the prefi symbols are lower case (e.g. m ) but a few are upper case (e.g. M )

32 Dr Roger Ni (Queen Mary, University of London) -.3 ) Prefies are written directly attached to the unit they modify, mg milligram 0 3 g whereas products of separate units are written with a space between them. Thus m g metre gram 3) The base unit for mass (kg) is unusual in that it contains a prefi. Fractions and multiples of it are formed by adding the appropriate prefi to the word gram and the symbol g. Thus mg not µkg and Mg not kkg Representation of Physical Quantities In general, Physical Quantity umerical Value Unit e.g. Mass 760 kg which is normally just written as Mass 760 kg Without its units, a physical quantity is meaningless (unless it happens to be dimensionless). It is conventional to leave a small space between the numerical value and the unit. e.g. Length m not m In calculations, you should normally convert all quantities into their standard SI units (without prefies) before entering them into the equation. For a well-behaved equation the answer will then also be in the appropriate SI units. However, if you are uncertain, you can also derive (check) the units for the quantity to be calculated using the relevant equation at the same time as you perform the numerical calculation, and always give the units as well as a numerical value for your answer. e.g. Density mass / volume ( ρ m / V ) If : m 36 kg, V 0.43 m 3 - what is the density? m ρ V 36 kg 0.43 m kg m kg m 3 3 e.g. Kinetic Energy ½ mass velocity ( E k ½ m v ) If : m kg, v m s - what is the kinetic energy? E (.04 0 ) k ( kg) (.04 0 m s ) kg (m s ) E k 6 kg m s J

33 Amount of Substance and Molecular Masses The Mole Dr Roger Ni (Queen Mary, University of London) -.33 mole (SI symbol mol ) of a substance contains as many specified elementary entities (atoms, molecules, etc ) as there are atoms in eactly g of the pure C isotope of carbon. Avogadro Constant The number of elementary entities in a mole is the Avogadro constant, A (or L ) mol. Eample 3 moles (3 mol) of hydrogen ( H molecules) contain H molecules. 3 moles (3 mol) of hydrogen atoms contain H atoms. Masses from Moles and Moles from Masses The molar mass of substance X is conventionally denoted M and given in the non-si units of g mol ; it is then numerically equal to the relative molecular mass (RMM, M r ) of X. The RMM is the sum of the relative atomic masses (RAM) of the atoms in the molecule where the values of the RMM and RAM are based on the assumption that all isotopes of elements are present in their natural abundances. An amount, n moles, of substance X will have a mass m (in g ) where m n M [ Check: units of m mol g mol - g ] Eample.0 mol of sodium has a mass of.0 mol 3.0 g mol - 46 g Conversely, a mass m (in g ) of substance X contains n moles, where n m M g [ Check: units of n mol - - g mol mol ] Eample.6 g.6 g of oygen ( O molecules) contain - 3 g mol mol of dioygen molecules. Masses of Individual Molecules A molecular mass (m) is specific to one molecule and dependent upon the actual isotopes of the elements present in the molecule thus the molecular mass of C 6 O is different from 3 C 6 O.

34 Dr Roger Ni (Queen Mary, University of London) -.34 Calculation of molecular masses from molar masses If, and only if, a substance is isotopically pure (e.g. F, which is 00% 9 F ) then the mass of one molecule of X (the molecular mass m ) is given by m X molar mass of X per mole number of molecules of X per mole M M m X A mol where m will have units of g if M r has its conventional units of g mol. To get the mass m in kg it will be necessary to further divide the value obtained by 000 in order to convert g to kg. Calculation of molecular masses using the atomic mass unit A more reliable method of calculating an accurate value of molecular mass uses the values of the isotopic masses epressed in atomic mass units. The atomic mass unit (u) is defined to be / th of the mass of one atom of C, i.e. m( C) u and has a value of u kg (to 6 s.f.) The molecular mass is then simply given by: m ( isotopic masses of atoms) atoms where it is necessary to use accurate, tabulated values for the isotopic masses. Eample The molecular mass of a H 35 Cl molecule is: m( H 35 Cl) m( H) m( 35 Cl).0078 u u u ( kg ) kg If the molar mass of pure H 35 Cl is required then this can be obtained by multiplying by the Avogadro constant. Mass of one mole of H 35 Cl kg mol kg mol g mol - (This is about 0.5 g lower than the molar mass value for normal HCl, which contains around 5% of H 37 Cl )