Physical Chemistry - Problem Drill 02: Math Review for Physical Chemistry

Transcription

1 Physical Chemistry - Problem Drill 02: Math Review for Physical Chemistry No. 1 of The Common Logarithm is based on the powers of 10. Solve the logarithmic equation: log(x+2) log(x-1) = 1 (A) 1 (B) 10 (C) 1/2 (D) 4/3 (E) 2 Good try. Set the same base on both side log b M = log b N and then set M = N. Solve x. Good try. Set the same base on both side log b M = log b N and then set M = N. Solve x. Good try. Set the same base on both side log b M = log b N and then set M = N. Solve x. D. Correct! Good job. Set the same base on both side log b M = log b N and then set M = N. Solve x. Good try. Set the same base on both side log b M = log b N and then set M = N. Solve x. Correct Answer: D To solve the logarithmic equation, set the same base on each side and equate the arguments to each other and solve: log b M = log b N M = N log(x+2) log(x-1) = 1 log[(x+2)/(x-1)] = log10 (x+2)/(x-1) = 10 x + 2 = 10x 10 9x = 12 x = 4/3

2 No. 2 of Natural Logarithms are based on powers of e. Calculate ½(4ln2-2ln5). (A) e (B) 1 (C) ln(4/5) (D) ln4 (E) ln10 Good try. Apply the basic rules of logarithm and computer the expression. Good try. Apply the basic rules of logarithm and computer the expression. C. Correct! Good job. Apply the basic rules of logarithm and computer the expression, ½(4ln2-2ln5) = ½(ln2 4 -ln5 2 ). Good try. Apply the basic rules of logarithm and computer the expression. Good try. Apply the basic rules of logarithm and computer the expression. Correct Answer: C Use the logarithm s power rule and quotient rule. ½(4ln2-2ln5) = ½(ln2 4 -ln5 2 ) = ½ln(16/25) = ln(16/25) ½ = ln(16/25) ½ = ln(4 2 /5 2 ) ½ = ln(4/5)

3 No. 3 of Subtracting logarithms of values yields the logarithm of the quotient of the values. Find x in the logarithmic equation logx log(x-2) = log2. (A) 1 (B) 4 (C) 2 (D) 10 (E) 100 Good try. Hint: Apply the quotient rule and convert the arguments on both sides equal. Solve for x. B. Correct! Good work. Apply the quotient rule and convert the arguments on both sides equal. Solve for x. Good try. Hint: Apply the quotient rule and convert the arguments on both sides equal. Solve for x. Good try. Hint: Apply the quotient rule and convert the arguments on both sides equal. Solve for x. Good try. Hint: Apply the quotient rule and convert the arguments on both sides equal. Solve for x. Correct Answer: B They are already with the same base. Apply the quotient rule and convert the arguments on both sides equal. Solve for x. logx log(x-2) = 100 log[x/(x-2)] = log2 x/(x-2) = 2 x = 4

4 No. 4 of Adding logarithms of a set of values yields the logarithm of the product of the values. Simplify and evaluate: ½log36 log15 + 2log5. (A) 10 (B) 2 (C) 1 (D) 3 (E) 0 Good try. Hint: Use both the product rule and quotient rule. Simplify the expression. Good try. Hint: Use both the product rule and quotient rule. Simplify the expression. C. Correct! Good job! Apply both the product rule and quotient rule. Simplify the expression. Good try. Hint: Use both the product rule and quotient rule. Simplify the expression. Good try. Hint: Use both the product rule and quotient rule. Simplify the expression. Correct Answer: C Apply the product rule and quotient rule: ½log36 log15 + 2log5 = log36 ½ log15 + log5 2 = log(6x25/15) = log10 = 1

5 No. 5 of Multiplying a logarithm by a value yields the logarithm of the power or root of the value. Solve the logarithmic equation: ½logx 4 log(2x-1) = logx 2 + log2. (A) 1 (B) 2 (C) ½ (D) ¾ (E) 10 Good try. Hint: Apply the power rule and set both sides equal to each other with the same base. Solve for x. Good try. Hint: Apply the power rule and set both sides equal to each other with the same base. Solve for x. Good try. Hint: Apply the power rule and set both sides equal to each other with the same base. Solve for x. D. Correct! Good job! Apply the power rule and set both sides equal to each other with the same base. Solve for x. Good try. Hint: Apply the power rule and set both sides equal to each other with the same base. Solve for x. Correct Answer: D Apply the power rule and set both sides equal to each other with the same base. ½logx 4 log(2x-1) = logx 2 + log2 logx 2 log(2x-1) = logx 2 + log2 log[x 2 /(2x-1)] = log(2x 2 ) Set the arguments equal on both sides and remove the log function. x 2 /(2x-1) = 2x 2 1/(2x-1) = 2 x = ¾

6 No. 6 of The Complex Conjugate of a Complex number is formed by reversing the sign of the imaginary part of the complex number. What are the complex conjugates of - 3-4i and 5. (A) -3-4i and - 5 (B) 3+4i and 5 (C) -3-4i and i 5 (D) 4i and 5i (E) -3+4i and 5 Good try. Hint: Think about x+yi s complex conjugate (x-yi). The real number s conjugate is always itself. Good try. Hint: Think about x+yi s complex conjugate (x-yi). The real number s conjugate is always itself. Good try. Hint: Think about x+yi s complex conjugate (x-yi). The real number s conjugate is always itself. Good try. Hint: Think about x+yi s complex conjugate (x-yi). The real number s conjugate is always itself. E. Correct! Good job. Recall x+yi s complex conjugate (x-yi). The real number s conjugate is always itself. Correct Answer: E Think about x+yi s complex conjugate (x-yi). The real number s conjugate is always itself (lack of imaginary part). -3-4i -3+4i (conjugate pair) 5 5 (conjugate pair)

7 No. 7 of A Vector is a quantity that has both. (A) A numerator and a denominator. (B) A square and a square root. (C) A quotient and a product. (D) A sum and a difference. (E) Direction and magnitude. Good try. Hint: Think about the properties of an arrow in space. Good try. Hint: Think about the properties of an arrow in space. Good try. Hint: Think about the properties of an arrow in space. Good try. Hint: Think about the properties of an arrow in space. E. Correct! Good work. A Vector is a quantity that has both direction and magnitude. Correct Answer: E A Vector is a quantity that has both direction and magnitude, as determining the position of one point in space relative to another.

8 No. 8 of Unit Vectors i, j, and k have a magnitude of and convert magnitudes into vector components along their respective directions: x, y, and z. (A) 10 units (B) 100 units (C) 1000 units (D) units (E) 1 unit Good try. Hint: Think about the properties of the number 1. Good try. Hint: Think about the properties of the number 1. Good try. Hint: Think about the properties of the number 1. Good try. Hint: Think about the properties of the number 1. E. Correct! Good work. Unit Vectors i, j, and k have a magnitude of 1 and convert magnitudes into vector components along their respective directions: x, y, and z. Correct Answer: E Unit Vectors i, j, and k have a magnitude of 1 and convert magnitudes into vector components along their respective directions: x, y, and z. For example, the unit vector for a vector v(2i + 4j k) is (2/ 21)i + (4/ 21)j + (-1/ 21)k where v = [ (-1) 2 ) = 21 (the magnitude of the original vector).

9 No. 9 of Vectors may be added by drawing them as arrows with appropriate direction and lengths, representing magnitude, and connecting the drawings. (A) Head to head. (B) Tail to tail. (C) Head to tail. (D) All of the above are correct. (E) None of the above is correct. Good try. Think about the relationship between an arrow and a vector. Good try. Hint: Think about the relationship between an arrow and a vector. C. Correct! Good work. Vectors may be added by drawing them as arrows with appropriate direction and lengths, representing magnitude, and connecting the drawings head to tail. Good try. Hint: Think about the relationship between an arrow and a vector. Good try. Think about the relationship between an arrow and a vector. Correct Answer: C Vectors may be added by drawing them as arrows with appropriate direction and lengths, representing magnitude, and connecting the drawings head to tail.

10 No. 10 of An infinitesimal change in a function or variable is a. (A) Differential. (B) Derivative. (C) Slope. (D) Sum. (E) None of the above is correct. A. Correct! Good work. An infinitesimal change in a function or variable is a differential. Good try. Hint: Think about the meaning of a limit. Good try. Hint: Think about the meaning of a limit. Good try. Hint: Think about the meaning of a limit. Good try. Hint: Think about the meaning of a limit. Correct Answer: A An infinitesimal change in a function or variable is a differential. Differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus.