How much heat does it take to increase the temperature of 1.80 mol of an ideal gas by 50.0 K near room temperature if the gas is held at constant volume and is (a) diatomic; (b) monatomic?

Given data;

Number of moles of ideal gas is n = 1.8 mol

Temperature deference is $∆T=50K$

Gas constant is R = 8.314J/mol K

Formula;

Equation of heat

role="math" localid="1665127031189" $Q=n{C}_v\Delta T$ ................. (1)

Molar heat capacity of diatomic molecules like H₂gas is

Molar heat capacity of monatomic molecules like H gas is

Put the value of molar heat capacity of diatomic molecules in equation (1)

$$\begin{gathered} \mathrm{Q}=\mathrm{n}\frac{5}{2}\mathrm{R}∆\mathrm{T} \\ =1.8\mathrm{mol}\times \frac{5}{2}\times 8.314\mathrm{J}/\mathrm{mol}\mathrm{K}\times 50\mathrm{K} \\ =1870.65\mathrm{J} \end{gathered}$$

Hence, the heat required to increase the temperature of diatomic is 1870.65J

Put the value of molar heat capacity of monatomic molecules in equation (1)

$Q=n\frac{3}{2}R∆T$

$$\begin{gathered} =1.8mol\times \frac{3}{2}\times 8.314J/molK\times 50K \\ =1122.39J \end{gathered}$$

Hence, the heat required to increase the temperature of monatomic is 1122.39J.