Chapter 13, Problem 81GP

(a)

To determine

To find:The factor at which the pressure changes.

Answer to Problem 81GP

The factor at which the pressure changes is $1.65$ .

Explanation of Solution

Given:

Initial Temperature, $T_1=110°\text{C = }383\text{ K}$

Final temperature, $T_2=360°\text{C = }633\text{ K}$

Formula used:

Ideal gas law formula is applied for the constant volume and constant number of moles.

  $P\times V=n\times R\times T$

Here, $P$ is the pressure,

  $V$ is the volume,

  $n$ is the number of molecules,

  $R$ is the universal gas constant,

  $T$ is the Temperature.

Calculation:

Substitute the given values in the above equation,

  $P_1\times V=n\times R\times T_1$   ... Equation (1)

  $P_2\times V=n\times R\times T_2$  ... Equation (2)

Taking ratio of the above equations,

  $\begin{array}{l}\frac{P_1\times V}{P_2\times V}=\frac{n\times R\times T_1}{n\times R\times T_2}\\ \frac{P_1}{P_2}=\frac{T_1}{T_2}=\frac{633}{383}=1.65\end{array}$

Conclusion:

Thus, the factor at which the pressure changes is $1.65$ .

(b)

To determine

To find:The factor at which $v_{rms}$ changes.

Answer to Problem 81GP

The factor at which $v_{rms}$ changes is $1.29$ .

Explanation of Solution

Given:

Initial temperature, $T_1=110°\text{C = }383\text{ K}$ .

Increase in temperature, $T_2=360°\text{C = }633\text{ K}$ .

Formula used:

The factor of root mean square velocity v_rms for constant mass is expressed by following formula,

  $v_{rms}=\sqrt{\frac{3\times k\times T}{m}}$

  $T$ is the temperature of gas.

  $m$ is the mass of the gas.

  $k$ is the Boltzmann constant.

Calculation:

Substitute the given values in the above equation,

  $v_{rm{s}_1}=\sqrt{\frac{3\times k\times T_1}{m}}$…. Equation (1)

  $v_{rm{s}_2}=\sqrt{\frac{3\times k\times T_2}{m}}$…. Equation (2)

Taking ratio of the above equations,

  $\frac{v_{rm{s}_2}}{v_{rm{s}_1}}=\frac{\sqrt{\frac{3\times k\times T_2}{m}}}{\sqrt{\frac{3\times k\times T_1}{m}}}=\sqrt{\frac{T_2}{T_1}}=\sqrt{\frac{633}{383}}=1.29$

Conclusion:

Thus,the factor at which $v_{rms}$ changes is $1.29$ .