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1 UNIVERSITI MALAYA UNIVERSITY OF MALAYA PEPERIKSAAN IJAZAH SARJANA MUDA SAINS EXAMINATION FOR THE DEGREE OF BACHELOR OF SCIENCE PEPERIKSAAN IJAZAH SARJANA MUDA SAINS DENGAN PENDIDIKAN EXAMINATION FOR THE DEGREE OF BACHELOR OF SCIENCE WITH EDUCATION SESI AKADEMIK 2015/2016 : SEMESTER I ACADEMIC SESSION 2015/2016 : SEMESTER I SCES1101 : MATEMATIK ASAS UNTUK KIMIA BASIC MATHEMATICS FOR CHEMISTRY Dis 2015/Jan 2016 Dec 2015/Jan 2016 Masa: 2 jam Time: 2 hours ARAHAN KEPADA CALON: INSTRUCTION TO CANDIDATES: Jawab SEMUA soalan. Answer ALL questions. (Kertas soalan ini mengandungi 9 soalan dalam 7 halaman yang dicetak) (This question paper consists of 9 questions on 7 printed pages)

2 1. Hamparan berikut memaparkan lima pentitratan yang dilakukan oleh seorang pelajar. Following spreadsheet displays five replicate titrations done by a student. A B C D E F G H 1 i x i x 2 x i /ml (x i x ) (a) Kirakan purata aritmetik, median dan sisihan piawai bagi nilai titer. Calculate the arithmetic mean, median and standard deviation of the titre values. Tentukan selang keyakinan 95% bagi x. Memandangkan Determine the 95% confidence interval for the x. Given that t υ=3 = 3.182, t ν=4 = 2.776, t ν=5 = & t υ=6 = (c) (d) (e) Tuliskan fungsi EXCEL bagi pengiraan x i dan x di G2 dan H2 dengan mengunakan rujukan sel yang sesuai. Write the EXCEL functions for the calculation of x i and x in G2 and H2 using appropriate cell references. Jika nilai x dikembalikan ke sel H2 dengan betul, tuliskan formula bagi pengiraan (x i x ) di B3. Formula ini mesti berupaya untuk mengembalikan nilai-nilai yang sesuai semasa disalin dari sel B3 sampai sel F3. If the value of x is returned to cell H2 correctly, write a formula for the calculation of (x i x ) in B3. This formula must able to return appropriate values when reproduced from cell B3 to cell F3. Apakah nilai jangkaan apabila jumlahkan nilai-nilai yang dikembali dari sel B3 ke sel F3? What is the expected value when total up the returned values from cell B3 to cell F3? (16 markah/marks) 2. Jika terdapat dua elektron untuk diagih kepada tiga orbital, p x, p y and p z. If there are two electrons to be distributed into three orbitals, p x, p y and p z. (a) Tentukan bilangan mikrokeadaan dalam konfigurasi p 2. Determine the number of microstates in a p 2 configuration. 2/7

3 Tanpa mempertimbangkan tenaga, cari kebarangkalian untuk mikrokeadaan di mana kedua-dua elektron tidak berada dalam orbital yang sama. Without considering energy, find the probability for the microstates where both electrons are not located in a same orbital. (6 markah/marks) 3. Dalam kimia kuantum, zarah biasanya digambarkan oleh fungsi gelombang kompleks: In quantum chemistry, a particle is generally described by a complex wavefunction: ψ(x) = a(x) + ib(x) di mana a(x) dan b(x) adalah fungsi yang berkaitan dengan zarah tersebut. where a(x) and b(x) are functions related to the particle. Tunjukkan bahawa taburan mod-kuasa dua: Show that the mod-squared distribution: ψ(x) 2 = a(x) 2 + b(x) 2 (5 markah/marks) 4. Dalam penghampiran Hückel, fungsi gelombang bagi π-orbital dalam molekul etena ialah kombinasi linear dua orbital atom φ 1 dan φ 2 In Hückel approximation, the wavefunction for the π-orbitals in ethene molecule is a linear combination of two atomic orbitals φ 1 and φ 2 ψ = c 1 φ 1 + c 2 φ 2 dan model tersebut boleh diringkaskan dalam bentuk matrik and the model can be simplify in matrix form α E β ( β α E ) (c 1 c ) = 0 2 di mana α, β, c 1, c 2 adalah parameter kepada model dan E adalah tenaga yang sepadan dengan orbital molekul. where α, β, c 1, c 2 are parameters of the model and E is the energy corresponding to the molecular orbital. (a) Tuliskan penentu sekular Hückel bagi etena Write the Hückel secular determinant for ethene 3/7

4 Tunjukkan aras tenaga terkuantum ialah Show that the quantized energy levels are E 1 = α + β and E 2 = α β (c) Tentukan jurang tenaga yang terlibat Determine the energy gap (d) Tunjukkan fungsi gelombang yang sepadan dengan E 1 yang mempunyai komponen c 1 dan c 2, mesti mematuhi Show that the wavefunction corresponding to E 1 which has components c 1 and c 2, must satisfy c 1 = c 2 (e) Tunjukkan fungsi gelombang ternormal yang sepadan dengan E 1 ialah Show that the normalized wavefunction corresponding to E 1 is ψ 1 = 1 2 (φ 1 + φ 2 ) (16 markah/marks) 5. Pertimbangkan molekul fosgen dengan orientasi berikut: Consider a phosgene molecule with following orientation: y Cl Cl C x O Anggapkan ia adalah spesies satah trigonal dengan semua sudut ikatan 120, magnitud momen dwikutub bagi setiap ikatan C Cl adalah 1.5 D dan 2.5 D untuk C=O. Tentukan momen dwikutub keseluruhan dalam unit vektor Cartesian. Assuming that it was a trigonal planar species with all bond angles of 120, the magnitude of dipole moment associated with each of the C Cl bond was 1.5 D and 2.5 D for C=O. Determine the overall dipole moment in Cartesian based unit vector. (7 markah/marks) 4/7

5 6. (a) Jika suatu bahan 2.34 g pada suhu 22 C dengan muatan haba tentu 3.88 kalori/g C dipanaskan dengan 124 kalori tenaga, apakah suhu baru bahan tersebut? Diberi If 2.34 g of a substance at 22 C with a specific heat of 3.88 cal/g C is heated with 124 calorie of energy, what is the new temperature of the substance? Given that C = q m T di mana C ialah muatan haba tentu yang berunit kalori/g C, q ialah haba yang ditambah dan berunit kalori, m ialah jisim yang berunit gram, dan T ialah kenaikan suhu pada bahan yang berunit C. where C is the specific heat in cal/g C, q is the heat added in calories, m is mass in gram, and T is the rise in temperature of the material in C. Menyusun semula persamaan ph, Rearrange the ph equation, ph = log 10 [H + ] supaya [H + ] ialah subjek formula. so that [H + ] is the subject of the formula. (10 markah/marks) 7. (a) Taburan Maxwell Boltzmann The Maxwell Boltzmann Distribution f(v) = Av 2 e Bv2 ialah satu taburan kebarangkalian untuk mencari kelajuan zarah tertentu v dalam 3 dimensi. A dan B ialah pemalar positif. is a probability distribution of finding particles at certain speed v in 3 dimensions. A and B are positive constants. (i) Cari pembezaan Find the differential d dv f(v) (ii) Carikan ungkapan untuk kelajuan yang paling mungkin? Find the expression for the most probable speed? 5/7

6 Pertimbangkan persamaan, Consider the equation, H = U + pv Bezakan persamaan ini terhadap suhu, T, pada isipadu tetap, V. p ialah tekanan dan U ialah tenaga dalaman. Differentiate this equation with respect to temperature, T, at constant volume, V. p is pressure and U is internal energy. 8. (a) Nilaikan kamiran Evaluate the integrals (15 markah/marks) (i) (ii) ln x x 2 dx π 2 sin x cos 3 x dx 0 (u = cos x) Dalam elektrokimia, perubahan arus elektrik, I, pada suatu masa, t selepas elektrod direndam dalam suatu larutan dapat dihitung melalui persamaan Cottrell In electrochemistry, the change of electric current,i, at a time, t after an electrode is immersed in a solution is calculated by the Cottrell equation I = nfac D πt di mana D adalah pekali resapan, c ialah kepekatan larutan, A ialah luas permukaan elektrod, F adalah pemalar Faraday, dan n adalah bilangan elektron yang dipindahkan. where D is the diffusion coefficient, c is the concentration in bulk solution, A is the surface area of the electrode, F is Faraday s constant, and n is the number of electrons transferred. Arus ditakrifkan sebagai kadar perubahan cas, Current is defined as the rate change of charge, I = dq dt 6/7

7 Dapatkan ungkapan bagi cas, Q. Find an expression for charge, Q. (15 markah/marks) 9. (a) Cari penyelesaian am bagi persamaan pembezaan Find the general solution of the differential equation y 8y + 16y = 0 Apakah sebutan keempat dalam pengembangan Taylor untuk What is the fourth term in the Taylor expansion for f(x) = xe x apabila a = 0? When a = 0? (10 markah/marks) TAMAT END 7/7

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