Chapter 13.1, Problem 13PP

Interpretation Introduction

Interpretation: The volume of the gas is to be calculated to know whether the piston will move up or down.

Concept introduction:

The ideal gas equation helps to calculate the volume of a gas at a given temperature and pressure. According to this law,

  $\mathrm{PV}=nRT$

Where,

  $\mathrm{P}$ is the pressure of the gas

  $\mathrm{V}$ is the volume of the gas

  $\mathrm{n}$ is the number of moles of the gas

  $\mathrm{R}$ is the ideal gas constant

  $\mathrm{T}$ is the temperature of the gas.

Answer to Problem 13PP

The piston will move down.

Explanation of Solution

Given information:

The initial pressure of the gas $P_1=1atm$

The initial volume of the gas $V_1=30mL$

The initial temperature of the gas $T_1=0C$

The final pressure of the gas $P_2=1.20atm$

The final temperature of the gas ${\mathrm{T}}_2=30°\mathrm{C}$

The ideal gas equation helps to calculate the volume of a gas at a given temperature and pressure. According to this law,

  $\mathrm{PV}=nRT$

Or

  $\frac{P_1{V}_1}{T_1}=\frac{P_2{V}_2}{T_2}$

Where

  ${\mathrm{P}}_1$ is the initial pressure of the gas

  ${\mathrm{V}}_1$ is the initial volume of the gas

  ${\mathrm{T}}_1$ is the initial temperature of the gas

  ${\mathrm{P}}_2$ is the final pressure of the gas

  ${\mathrm{V}}_2$ is the final volume of the gas

  ${\mathrm{T}}_2$ is the final temperature of the gas

Now converting the Celsius into kelvin.

  $\begin{array}{l}{T}_1(K)=T_1(C)+273\end{array}$

  $\begin{array}{c}=0+273\end{array}$

  $\begin{array}{c}=273K\end{array}$

  $\begin{array}{c}{T}_2=30C\end{array}$

  $\begin{array}{c}=30+273\end{array}$

  $\begin{array}{c}=303K\end{array}$

Now substituting the values into the formula,

  $\begin{array}{c}{V}_2=\frac{P_1{V}_1{T}_2}{P_2{T}_1}\end{array}$

   $\begin{array}{c}=\frac{1\times 30\times 303}{1.20\times 273}\end{array}$

   $\begin{array}{c}=27.75mL\end{array}$

Therefore, the volume of the gas is reduced to $27.75mL$ .

Conclusion

As the volume of gas is reduced to $27.75mL$ , the piston of the cylinder will move down.