# Changing systems with heat

### Heat Capacities

Heat capacity: (limiting ratio of) heat supplied, $\Delta Q$, divided by the change in temperature, $\Delta T$. $$C = \frac{\delta Q}{dT}.$$

• $\Delta Q$ is an extensive quantity: Usually we'll look instead at the specific heat capacity per kilomole.
• We haven't specified a path, but we know the $\delta Q$ is not an exact differential.

The two kinds of paths we'll consider are: $$c_v \equiv \frac{1}{n} $\frac{\delta Q}{dT}$_v = $\frac{\delta q}{dT}$_v \ ;\ c_P \equiv $\frac{\delta q}{dT}$_P.$$

Now, consider the internal energy function $u$:

• $u$ is a state function,
• $\Rightarrow$ specifying any 2 thermodynamic parameters is enough to uniquely determine the state and any function of the state.
• Your choice! $u(P,v)$, or $u(T,v)$, or $u(T,P)$.

Choosing $u(T,v)$: $$du = $\frac{\partial u}{\partial T}$_v dT + $\frac{\partial u}{\partial v}$_T dv.$$

For a constant volume process, the second term vanishes, leaving: $$$du$_v = $\frac{\partial u}{\partial T}$_v dT.$$

According to the 1st Law, $du=\delta q -\delta w$. For a reversible process $\delta w = P\,dv$: $$du = \delta q-P\,dv \Rightarrow $du$_v = \delta q.$$

Taking the last two expressions for $du$ in constant-volume processes, gives $$\delta q = $\frac{\partial u}{\partial T}$_v dT.$$

Or, $$$\frac{\delta q}{\partial T}$_v = $\frac{\partial u}{\partial T}$_v.$$

So,

$$c_v = $\frac{\partial u}{\partial T}$_v.$$

We shall shortly see that, for an ideal gas, the internal energy actually depends only on temperature, i.e. $u(T,v) = u(T)$, not on volume at all. When this is true, then we can axe the subscript from the partial derivative, and also dispense with the partial derivatives, and just write: $$c_v = \frac{du}{dT} \text{ (ideal gas)}.$$

We can integrate to find the overall energy in a relatively easy fashion: $$\int du = u-u_0 = \int_{T_0}^{T} c_v dT.$$

Problem (Carter 4-6 a): Pretend Oxygen is an ideal gas with $c_v=(5/2)R$. Suppose that the temperature of 2 kilomoles of O${}_2$ is raised from 27 C to 227 C. What is the increase in internal energy?

Problem (Carter 4-1): The specific heat capacity $c_v$ of solids at low temperature is given by the Debye T${}^3$ law: $$c_v = A $\frac{T}{\theta}$^3.$$

• $A$ is a constant where $A=19.4 \times 10^5$ J kilomole${}^{-1}$ K${}^{-1}$,
• $\theta$ is called the Debye temperature. For NaCl $\theta = 320$K.

Anne Petersen