Question

To what temperature must a neon gas sample be heated to double its pressure if the initial volume of gas at $75°\mathrm{C}$ is decreased by $15.0\%$ by cooling the gas?

Answer 1

Step 1: Given information

• Initial temperature, ${\mathrm{T}}_1=75°\mathrm{C}$
• Initial pressure, ${\mathrm{P}}_1$.
• Final pressure ${\mathrm{P}}_2=2{\mathrm{P}}_1$
• Initial volume $={\mathrm{V}}_1$
• The final volume is $15.0\%$ of the initial volume.

Step 2: The formula used

• $\frac{{\mathrm{P}}_1{\mathrm{V}}_1}{{\mathrm{T}}_1}=\frac{{\mathrm{P}}_2{\mathrm{V}}_2}{{\mathrm{T}}_2}$
• ${\mathrm{P}}_1$ is the initial pressure.
• ${\mathrm{V}}_1$ is the initial volume.
• ${\mathrm{P}}_2$ is the final pressure.
• ${\mathrm{V}}_2$ is the final volume.
• ${\mathrm{T}}_1$ is the initial temperature.
• ${\mathrm{T}}_2$ is the final temperature.

Step 2: Calculate the final volume

• The final volume is as follows:

$\begin{array}{rcl}{\mathrm{V}}_2& =& {\mathrm{V}}_1-\frac{15{\mathrm{V}}_1}{100}\\ & =& 0.85{\mathrm{V}}_1\end{array}$

Step 4: Convert the temperature from degree Celsius into Kelvin

• The conversion of degrees Celsius into Kelvin is as follows:

$\begin{array}{rcl}\mathrm{T}\left(\mathrm{K}\right)& =& \mathrm{T}\left(°\mathrm{C}\right)+273\\ {\mathrm{T}}_1& =& 75+273\\ & =& 348\mathrm{K}\end{array}$

Step 5: Calculate the temperature

• The final temperature is as follows:

$\begin{array}{rcl}\frac{{\mathrm{P}}_1{\mathrm{V}}_1}{{\mathrm{T}}_1}& =& \frac{{\mathrm{P}}_2{\mathrm{V}}_2}{{\mathrm{T}}_2}\\ {\mathrm{T}}_2& =& \frac{{\mathrm{P}}_2{\mathrm{V}}_2{\mathrm{T}}_1}{{\mathrm{P}}_1{\mathrm{V}}_1}\\ & =& \frac{2\bcancel{{\mathrm{P}}_1}\times 0.85\bcancel{{\mathrm{V}}_1}\times 348\mathrm{K}}{\bcancel{{\mathrm{P}}_1}\bcancel{{\mathrm{V}}_1}}\\ & =& 591.6\mathrm{K}\end{array}$

Therefore, the temperature at which a neon gas sample is heated to double its pressure is $\mathbf{591}\mathbf{.}\mathbf{6}\mathbf{}\mathbf{K}$.