homework1 - CHE 301 - Physical Chemistry I - Fall 2010...

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Unformatted text preview: CHE 301 - Physical Chemistry I - Fall 2010 homework1 Dust off your calc books and try to solve the following (read the Appendix of the text book as well) 1) Find df dx for the following by using the chain rule (a) f = (x2 − 2)−1 (b) f = ln x e−x 2 (c) f = sin3 x cos x 2) Integrate the following equations from x = a to x =b (a) f = 1/(1 + x) (b) f = sin 2x (c) f = 5x5 + x 3) Find ∂f ∂x and ∂f ∂y and df (the total differential) for the following: (a) f (x, y ) = xy − y − x (b) f (x, y ) = 2x2 − 2xy + y 2 (c) f (x, y ) = y − x sin y + 1 4) Consider the relation z = a x cos ( y ) b between the variables x, y , and z . The quantities a and b are constants. As written, the above equation expresses z as a function of the two variables x and y . (a) Calculate the partial derivatives i) ∂z ∂x ; y 1 ii) ∂z ∂y . x (b) Calculate the partial derivatives ∂x ∂y iii) ; ∂y ∂z iv ) z . x (c) Verify that your results in (a) and (b) conform to the cyclic rule ∂z ∂x ∂x ∂y y ∂y ∂z z = −1 . x 5) An expression like A(x, y ) dx + B (x, y ) dy is said to be an exact differential if the functions A(x, y ) and B (x, y ) satisfy the relation ∂A ∂y ∂B ∂x = x . y Determine whether the expressions that follow are exact differentials (a, b, and R are constants). (i) 3a x2 y dx + ( ax3 + 2by ) dy (ii) y dx (iii) y dx + x dy (iv) −(RT /p2 ) dp + (R/p) dT 6) For the following function xy 2 z 3 w3 z 3 − F (w, x, y, z ) = 3xy + 32y w 2 evaluate the following derivatives: a) ∂F ∂x ; b) w,y,z ∂F ∂w 2 ; c) x,y,z ∂F ∂y x,z,w d) ∂ ∂z ∂F ∂x ; e) w,y,z w,x,y ∂ ∂x ∂F ∂z w,x,y w,y,z ∂ ; f) ∂w ∂ ∂z ∂F ∂x w,y,z w,x,y 7) Determine the results for the following derivatives, assuming that i) the ideal gas law pV = nRT holds: ii) the Van de Waals gas law (p + an2 /V 2 )(V − nb) = nRT holds: a) ∂V ∂p ; b) T ,n ∂V ∂n ; c) T ,p ∂T ∂V 3 ; d) n,p ∂p ∂T ; e) n,V ∂p ∂n . T ,V x,y,z . ...
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homework1 - CHE 301 - Physical Chemistry I - Fall 2010...

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