91Ó°ÊÓ

• Heat added to the ideal gas,Q=20.9J
• Initial volume of the gas,V_i=50 cm³
• Final volume of the gas,V_f=100 cm³
• Pressure of the ideal gas,$P=1.00atmor1.01×{10}^{5}Pa$
• Quantity of the gas,$n=2.00×{10}^{-3}mol$

The expression for the amount of energy transferred to the gas is given by,

$Q=n{C}_P\mathrm{Δ}T$

Here Q is the amount of energy transferred to gas, n is the number of moles, C_Vis the specific heat capacity at constant volume and$\mathrm{Δ}T$is the temperature difference.

The expression for the work done by the gas is given by,

$W=p\mathrm{Δ}V$

Here W is the work done by the gas, p is the pressure and V is the change in the volume.

Formulae:

The first law of thermodynamics, $Q=E_{int}+W$ (i)

The ideal gas equation, PV=nRT (ii)

The heat absorbed by the body at constant pressure, $Q=n{C}_p\mathrm{Δ}\mathrm{T}$ (iii)

The relation of the molar specific heats at constant pressure and volume,

$C_p=C_v+R$ (iv)

The work done by a body at constant pressure,$W=P\mathrm{Δ}V$ (v)

Using equation (v) in equation (i) with the given data, the change in the internal energy of the ideal gas can be given as follows:

$$\begin{gathered} E_{int}=Q-P(V_f-V_i) \\ =20.9J-(1.01×{10}^{5}Pa)(100c{m}^3-50c{m}^3)\left(\frac{{10}^{-6}{m}^3}{1c{m}^3}\right) \\ =15.9J \end{gathered}$$

Hence, the value of internal energy is 15.9J.

Substituting the value of change in temperature from equation (ii) in equation (iii) with the given data, the specific heat at constant pressure can be given as follows:

$$\begin{gathered} C_P=\frac{Q}{n(P\mathrm{Δ}V/nR)} \\ =\frac{R}{P}\left(\frac{Q}{\mathrm{Δ}V}\right) \end{gathered}$$
$$\begin{gathered} =\frac{(8.314J/mol.K)(20.9J)}{(1.01×{10}^{5}Pa)\left[\left(100c{m}^3-50c{m}^3\right)\left({\displaystyle \frac{{10}^{-6}{m}^3}{1c{m}^3}}\right)\right]} \\ =34.4J/mol.K \end{gathered}$$

Hence, the value of specific heat at constant pressure is 34.4J/mol.K .

Using the above value in equation (iv), the specific heat at constant volume of the gas cab ne calculated as follows:

$$\begin{gathered} \begin{array}{l}{c}_V=C_p-R\\ =34.4J/mol.K-8.314J/mol.K \\ =26.1J/mol.K\end{array} \end{gathered}$$

Hence, the value of the specific heat is 26.1J/mol.K