A gas of $1.0\times {10}^{20}$atoms or molecules has $1.0J$ of thermal energy. Its molar specific heat at constant pressure is $20.8J/molK$. What is the temperature of the gas?

The temperature of a gas determines the average translational kinetic energy of the molecules.

The molecules in a hot gas move faster than those in a cold gas; the mass is constant, but the kinetic energy, and hence the temperature, is higher due to the molecules' increased velocity.

$T$If the weather changes, by $∆T$, then the thermoelectric power for a gas changes by the equation $(20.30)$in the form

$\Delta E_{\mathrm{th}}=n{C}_{V}T$--------(1)

Where $C_v$is the molecule initial temperature at constant volume, and it has a relationship with the molar specific heat at constant pressure.$C_p$by

$C_V=C_p-R$

We solve equation $\left(1\right)$for $T$to be in the form

$T=\frac{\Delta E_{th}}{n\left(C_p-R\right)}$

$n$is the number of moles there are. Knowing how many molecules there are $N$, we can get the number of moles by using Avogadro's number as next

$n=\frac{N}{N_{\mathrm{A}}}=\frac{1\times {{10}^{20}}_{molecules}}{6.023\times {{10}^{23}}_{molecules}/mole}=1.66\times {10}^{-4}\mathrm{moles}$

Now substitute we get,

$T=\frac{\mathrm{\Delta }{E}_{\mathrm{th}}}{n\left(C_{\mathrm{P}}-R\right)}$

$=\frac{1\mathrm{J}}{\left(1.66\times {10}^{-4}\mathrm{moles}\right)(20.8\mathrm{J}/\mathrm{mol}\cdot \mathrm{K}-8.314\mathrm{J}/\mathrm{mol}\cdot \mathrm{K})}$

$=482K$