2 The first law 2.2 The first law 2.4 The pressure

2.3 Specific heats

(September 1, 2023)

The first law involves heat. The heat inflows are characterized by the specific heat, which is the amount heat required, ${\mathchar 22\mskip-12.0mu d}Q$ , per change in temperature $d T$ .

If the volume is held fixed, we have the specific heat at constant volume

C_{V}\equiv\left(\frac{{\mathchar 22\mskip-12.0mu d}Q}{dT}\right)_{V}

(2.8)

The subscript $V$ indicates that the volume is held fixed. If the pressure is held fixed, we have the specific heat at constant pressure

C_{p}\equiv\left(\frac{{\mathchar 22\mskip-12.0mu d}Q}{dT}\right)_{p}

(2.9)

Energy and $C_{V}$ :

If the volume is held fixed $dV=0$ and ${\mathchar 22\mskip-12.0mu d}W=0$ . The change in energy is

dU=\left(\frac{\partial U}{\partial T}\right)_{V}dT+\left(\frac{\partial U}{% \partial V}\right)_{T}\cancelto{0}{dV}

(2.10)

and so from the first law, $dU={\mathchar 22\mskip-12.0mu d}Q$ , and

\boxed{C_{V}=\left(\frac{\partial U}{\partial T}\right)_{V}}

(2.11)

For an diatomic gas $U=\frac{5}{2}N{k_{\scriptscriptstyle B}T}$ , and so

\displaystyle C_{V}=

\displaystyle\frac{5}{2}Nk_{B}

diatomic ideal gas (2.12)

The specific heat grows with the number of particles in the system. For this reason we often quote the specific heat for one mole of substance, $C_{V}^{\rm 1ml}$ . The specific for one mole of an ideal gas diatomic gas is for instance

\displaystyle C_{V}^{\rm 1ml}=

\displaystyle\frac{5}{2}N_{A}k_{B}=\frac{5}{2}R

diatomic ideal gas (2.13)

We will discuss the energy $U(T,V)$ later. For any ideal gas (e.g. dilute water vapor) the energy takes the form

\displaystyle U=

\displaystyle Ne_{0}(T)

any ideal gas (2.14)

The specific heat takes the form

\displaystyle C_{V}=

\displaystyle Ne_{0}^{\prime}(T)

any ideal gas (2.15)

Or for one mole of substance

\displaystyle C_{V}^{\rm 1ml}=R\left(\frac{1}{k_{B}}\frac{{\rm d}e_{0}}{{\rm d% }T}\right)

any ideal gas (2.16)

Relating $C_{p}$ and $C_{v}$ :

The specific heat at constant pressure $C_{p}$ is larger than $C_{V}$ , because some of the heat added (per degree of temperature change) is used by the gas to do work as it gas expands at fixed pressure. By figuring how much work is done, we will show shortly that for any ideal gas (not just diatomic or monoatomic)

\displaystyle C_{p}=

\displaystyle C_{V}+Nk_{B}

any ideal gas (2.17)

\displaystyle C_{p}^{\rm 1ml}=

\displaystyle C_{V}^{1\rm ml}+R

any ideal gas (2.18)

In general $C_{p}$ and $C_{V}$ are related. We will show much later using the second law

\displaystyle C_{p}=

\displaystyle C_{V}+\frac{VT\beta_{p}^{2}}{\kappa_{T}}

all substances (2.19)

Here $\beta_{p}$ is the volume expansion coefficient, and $\kappa_{T}$ is the isothermal compressibility – see the next section.

The ratio of specific heats is given a name

\gamma\equiv\frac{C_{p}}{C_{V}}

(2.20)

The factor $\gamma$ is close to unity in practice and often nearly constant. For instance for a diatomic gas where $C_{V}=\tfrac{5}{2}Nk_{B}$ , we find using the relation in Eq. (any ideal gas (2.17)), that $\gamma=7/5$ .