Chapter 13, Problem 84GP

To determine

To Find: The RMS speed of nitrogen molecules.

Answer to Problem 84GP

The rms speed of nitrogen molecules at given pressure and volume is $439\text{ m/s}$ .

Explanation of Solution

Given info:

Pressure $P=4.2\text{ atm }=4.2\times 1.013\times {10}^5{\text{ N/m}}^{\text{2}}$ .

Volume $V=7.6{\text{ m}}^{\text{3}}$

Boltzmann constant $k=1.38\times {10}^{-23}\text{ J/K}$ .

Mass of nitrogen Temperature $m=28\times 1.66\times {10}^{-27}=4.648\times {10}^{-26}\text{ kg}$ .

Formula used:

Write the expression for RMS speed.

  $v_{rms}=\sqrt{\frac{3kT}{m}}$

Write the ideal gas law.

  $\begin{array}{l}P\times V=n\times R\times T\\ T=\frac{P\times V}{n\times R}\end{array}$

Here,

  $P$=Pressure

  $V$ =volume

  $\eta$ =amount of substance

  $k$ =Ideal gas constant

  $T$ =temperature

Calculation:

Substitute the given values, calculate the temperature, and then find the RMS speed.

  $T=\frac{4.2\times 1.013\times {10}^5\times 7.6}{1800\times 8.314}=216.067\text{ K}$

Calculate the RMS speed.

  $v_{rms}=\sqrt{\frac{3\times 1.38\times {10}^{-23}\times 216.067}{4.648\times {10}^{-26}}}=439\text{ m/s}$

Conclusion:

Thus, the RMS speed of nitrogen molecules at given pressure and volume is $439\text{ m/s}$ .