Question

A sample of a monatomic ideal gas at $\mathbf{1}\mathbf{.}\mathbf{00}\mathit{a}\mathit{t}\mathit{m}$and ${\mathbf{25}}^{\mathbf{o}}\mathit{C}$expands adiabatically and reversibly from $\mathbf{5}\mathbf{.}\mathbf{00}\mathit{L}$to$\mathbf{12}\mathbf{.}\mathbf{5}\mathit{L}$ . Calculate the final temperature and pressure of the gas, the work associated with this process, and the change in internal energy.

Short answer

The final temperature is ${\text{P}}_{\text{2}}=0.218\text{atm}$

The final pressure is ${\text{T}}_{\text{2}}=163\text{K}$

Internal energy and work associated with the process $\text{ΔU}=-344\text{J}$

Step-by-step solution

Definition of adiabatic process

A process that happens without transferring heat or mass between a thermodynamic system and its surrounding environment is called an adiabatic process.

Calculations

Number of moles in mono atomic gas

$\begin{array}{l}\text{PV}=\text{nRT}\\ \text{n}=\frac{\text{PV}}{\text{RT}}\\ \text{n}=0.204\text{\hspace{0.17em}mol}\end{array}$

Monoatomic ideal gas is:

$\begin{array}{l}\gamma =1.66\\ {\text{P}}_{\text{2}}=0.218\\ {\text{T}}_{\text{2}}=163\text{K}\end{array}$

Calculation for${\text{C}}_{\text{v}}$

For monoatomic gas,

data-custom-editor="chemistry" $\begin{array}{l}\gamma =\frac{{\text{C}}_{\text{p}}}{{\text{C}}_{\text{v}}}\\ \gamma =\frac{5}{3}\\ {\text{C}}_{\text{p}}-{\text{C}}_{\text{v}}=\text{R}\\ {\text{C}}_{\text{v}}=\left(\frac{3}{2}\right)\text{R}\end{array}$

In an adiabatic process, heat change is zero.

$\begin{array}{l}\text{ΔU}=\text{q+W}\\ \text{ΔU}=\text{W}\\ \text{ΔU}=-344\text{J}\end{array}$