Chapter 5, Mathematical tools Video Solutions, Thermodynamics: A complete undergraduate course

Problem 1

(i) Consider the following small quantity: $y^2 \mathrm{~d} x+x y \mathrm{~d} y$. Find the integral of this quantity from $(x, y)=(0,0)$ to $(x, y)=(1,1)$, first along the path consisting of two straight-line portions $(0,0)$ to $(0,1)$ to $(1,1)$, and then along the diagonal line $x=y$. Comment.
(ii) Now consider the small quantity $y^2 \mathrm{~d} x+2 x y \mathrm{~d} y$. Find (by trial and error or any other method) a function $f$ of which this is the total differential.

Wendi Zhao

Numerade Educator

03:04

Problem 2

A certain small quantity is given by
$$
C_V \mathrm{~d} T+\frac{R T}{V} \mathrm{~d} V
$$
where $C_V$ and $R$ are constants and $T$ and $V$ are functions of state. (i) Show that this small quantity is an improper differential. (ii) Let $\mathrm{d} Q=C_V \mathrm{~d} T+$ $\frac{R T}{V} \mathrm{~d} V$. Show that $\mathrm{d} Q / T$ is a proper differential.

Lottie Adams

Numerade Educator

03:38

Problem 3

If $x, y$ are functions of state, give an argument to prove that
$$
\frac{\mathrm{?} Q_{(\mathrm{p})}}{\mathrm{d} x}=\left.\frac{\mathrm{?} Q_{(\mathrm{p})}}{\mathrm{d} y} \frac{\partial y}{\partial x}\right|_{(\mathrm{p})},
$$
where the subscript (p) indicates the path along which the changes are evaluated.

Charles Machakwa

Numerade Educator

01:04

Problem 4

$A$ and $B$ are both functions of two variables $x$ and $y$, and $A / B=C$. Show that
$$
\left.\frac{\partial x}{\partial y}\right|_C=\frac{\left.\frac{\partial \ln B}{\partial y}\right|_x-\left.\frac{\partial \ln A}{\partial y}\right|_x}{\left.\frac{\partial \ln A}{\partial x}\right|_y-\left.\frac{\partial \ln B}{\partial x}\right|_y}
$$
[Hint: develop the left-hand side, and don't forget that for any function $f,(\mathrm{~d} / \mathrm{d} x)(\ln f)=(1 / f) \mathrm{d} f / \mathrm{d} x]$.

Aman Gupta

Numerade Educator