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1 UNIVERSITI MALAYA UNIVERSITY OF MALAYA PEPERIKSAAN IJAZAH SARJANA MUDA SAINS EXAMINATION FOR THE DEGREE OF BACHELOR OF SCIENCE SESI AKADEMIK 2015/2016 : SEMESTER 2 ACADEMIC SESSION 2015/2016 : SEMESTER 2 SIC1005 : MATEMATIK ASAS UNTUK KIMIA BASIC MATHEMATICS FOR CHEMISTRY Jun 2016 June 2016 Masa : 2 jam Time : 2 hours ARAHAN KEPADA CALON: INSTRUCTION TO CANDIDATES: Jawab SEMUA soalan. Answer ALL questions. (Kertas soalan ini mengandungi 11 soalan dalam 6 halaman yang dicetak) (This question paper consists of 11 questions on 6 printed pages)

2 1. Hamparan berikut memaparkan ukuran zarah halus di udara oleh seorang ahli kimia atmosfera yang diperolehi daripada enam stesen pensampelan di sepanjang sebuah lebuhraya. The following spreadsheet displays the measurements of fine particulate matter in the air by an atmospheric chemist obtained from six sampling stations along a highway. A B C D E F G 1 Stesen / Station, i Ukuran / Measurement, x i ( g m -3 ) Hitungkan purata, median, sisihan piawai dan ralat piawai bagi ukuran tersebut. Calculate the mean, median, standard deviation and standard error of the measurements. Anggarkan selang keyakinan 95% bagi nilai purata. Diberi; Estimate the 95% confidence interval for the mean value. Given that; t ν=4 = 2.776,t υ=5 = 2.571,t υ=6 = 2.447&t υ=7 = Pertimbangkan satu sistem dengan bilangan N zarah yang dapat dibezakan dengan jumlah tenaga E. Jika terdapat k keadaan tenaga; Consider a system of N number of distinguishable particles with total energy E. If there are k available states with energies; ε 1, ε 2, ε k di mana n 1 zarah berada pada keadaan tenagaε 1, n 2 zarah pada keadaan tenaga ε 2, dan sebagainya. where n 1 particles in energy state ε 1, n 2 particles in energy state ε 2,and so on. Tentukan bilangan cara, Ω, untuk zarah tersebut dapat diagihkan di antara k keadaan zarah dalam sebutan N dan n i. Determine the number of ways,ω, of the particles can be distributed among k particle states in terms of N and n i. 2/6

3 Dengan Penghampiran Sterling, tunjukan bahawa With Sterling s Approximation, show that ln Ω = N ln N n i ln n i k i 3. Pemalar dielektrik untuk suatu larutan cair molekul berkutub yang diukur pada frekuensi ω bagi medan elektrik ulang-alih yang berbeza pada masa τ diberikan oleh fungsi kompleks berikut; The dielectric constant of a dilute solution of polar molecules measured at a frequency ω of an alternating electric field which varies over time τ is given by following complex function; ε = ε + iε = n 2 + N 0 ρμ 2 3Mε 0 k B T(1 + iωτ) di mana parameter n, N 0, ρ, μ, M, ε 0 dan k B adalah pemalar pada suhu T. where the parameters n, N 0, ρ, μ, M, ε 0 and k B are constants at temperature T. Tulis semula persamaan di atas untuk menentukan ε dan ε. Rewrite the above equation to determine ε and ε. 4. Persamaan matriks Huckel bagi siklopropena adalah The Hückel matrix equation for cyclopropeneis α E β β c 1 ( β α E β ) ( c 2 ) = 0 β β α E c 3 (10 markah/marks) di mana α, β, c 1, c 2 dan c 3 adalah parameter kepada model dan E adalah tenaga yang sepadan dengan orbital molekul. where α, β, c 1, c 2 and c 3 are parameters of the model and E is the energy corresponding to the molecular orbital. Untuk penyelesaian bukan-sifar, tuliskan penentu sekular Hückel dalam sebutan x, di mana For non-zero solution, write the Hückel secular determinant in terms of x, where x = α E β 3/6

4 Selesaikan penentuan secular tersebut dan tunjukkan bahawa tiga tenaga terkuantum ialah Solve the secular determinant and show that the three quantized energies are E 1 = α + β&e 2 = E 3 = α β (c) Tunjukkan bahawa vektor-eigen yang sepadan dengan E 1 mesti mempunyai Show that the eigen vector corresponding to E 1 must have c 1 = c 2 = c 3 (d) Tunjukkan bahawa vektor-eigen ternormal yang sepadan dengan E 1 ialah Show that the normalized eigen vector that corresponds to E 1 is c 1 = ( 1) 1 (18 markah/marks) 5. Penyerapan spektroskopi, A, bagi suatu sampel adalah berkadar dengan magnitud komponen dipole yang selari dengan arah polarisasi sinaran yang digunakan. Pertimbangkan sampel yang mempunyai momen dwikutub (μ) seperti di bawah: Spectroscopic absorbance, A, of a sample is proportional to the magnitude of the component of the dipole which is parallel to the direction of polarization of the radiation used. Consider a sample that has a dipole moment (μ) as below, y μ = 2.5D 31 x Tuliskan μ dalam sebutan vektor unit Cartesian i, j dan k. Write μ in terms of the Cartesian unit vectors i, j and k. 4/6

5 Kirakan nisbah penyerapan bagi sampel tersebut ketika sinaran adalah terpolarisasi secara menegak kepada ketika ia terpolarisasi secara mendatar. Calculate the ratio of absorbances of the sample when the radiation is vertically polarized to when it is horizontally polarized. (6 markah/marks) 6. Seorang ahli kimia industri menghasilkan dm 3 baja dalam satu tindak balas. An industrial chemist produces dm 3 of fertilizer in a reaction. Berapakah jumlah itu dalam m 3? How much is that in m 3? Jika tindak balas tersebut memerlukan tiga langkah. Langkah pertama memberikan hasil 66%, langkah kedua memberikan hasil 50% dan langkah ketiga memberikan hasil 95%. Berapakah peratus keseluruhan hasil sintesis tersebut? If the reaction requires 3 steps. The first step gives a 66% yield, the second step gives a 50% yield and the third step gives a 95% yield. What is the overall percentage yield for this synthesis? 7. Persamaan gas unggul adalah pv = nrt. Cari: The ideal gas equation is pv = nrt. Find: ( dv dt ) p ( dt dp ) V 8. Selesaikan persamaan pembezaan yang berikut untuk mendapatkan [A] sebagai satu fungsi t. Solve the following differential equation to find [A] as a function of t. v = d[a] dt = k 2[A] 2 5/6

6 9. Dalam hukum taburan Maxwell-Boltzmann, peluang untuk menjumpai satu partikel dengan kelajuan v (per unit isipadu) diberi sebagai; In the Maxwell-Boltzmann distribution, the likelihood of finding a particle with a particular velocity v (per unit volume) is given by; n(v) dv = 4πN V ( m 2πkT ) 3 2 v 2 e mv2 2kT dv Tunjukkan kelajuan yang paling berkemungkinan bagi satu partikel dalam hukum Maxwell-Boltzmann adalah; Show that the most probable speed for a particle in a Maxwell-Boltzmann distribution is; v mp = 2kT m 10. Persamaan van der Waals The van der Waals equation of states P = RT V b a V 2 Dapatkan jumlah pembezaan, dp, diberi a, b dan R adalah pemalar. Find the total differential,dp, given that a, b and R are constants. 11. Selesaikan Solve T f T f ] T 1 T 2 S = mc p [ dt T + dt T (10 markah/marks) TAMAT END 6/6

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