Chapter 13, Problem 75GP

To determine

To Find: The fraction of original gas.

Answer to Problem 75GP

The fraction of the original gas remaining in the cylinder is $20.69\text{ \% of }N$ .

Explanation of Solution

Given information:

  $\begin{array}{l}\text{Pressure }P=28\text{ atm }\Rightarrow P=1+28=29\text{ atm}\\ \text{Pressure }{P}^{\prime }=5\text{ atm }\Rightarrow P^{\prime }=1+5=6\text{ atm}\end{array}$

Here one is added because the gauge pressure is the difference between the absolute pressure and the atmospheric pressures.

Formula used:

  $\begin{array}{l}P\times V=N\times k\times T\\ \text{Fraction of original gas = }\frac{N^{\prime }}{N}\\ \frac{P\times V}{N\times T}=k=\frac{P^{\prime }\times V^{\prime }}{N^{\prime }\times T^{\prime }}\end{array}$

Calculation:

Pressure decreases after blown up the many balloons, due to a decrease in pressure now we have fewer molecules inside the cylinder. We want to determine the fraction of the original gas remaining in the cylinder, for that we can use the following formula.

  $\begin{array}{l}\frac{P\times V}{N\times T}=k=\frac{P^{\prime }\times V^{\prime }}{N^{\prime }\times T^{\prime }}\\ k\text{ is Boltzmann constant and removes while comparing}\text{.}\\ \frac{N^{\prime }}{N}=\frac{P^{\prime }}{P}\times \frac{T}{T^{\prime }}\times \frac{V^{\prime }}{V}\end{array}$

The volume of the cylinder is the same as it is a solid cylinder. Temperature is also constant because we are having free expansion inside the cylinder. Now, we will get

  $\frac{N^{\prime }}{N}=\frac{P^{\prime }}{P}$

Substitute the given information in the above equation,

  $\begin{array}{l}\frac{N^{\prime }}{N}=\frac{29}{6}=0.2069\\ N^{\prime }=0.2069\times N\\ N^{\prime }=20.69\text{ \% of }N\end{array}$

Conclusion:

Thus, the fraction of the original gas remaining in the cylinder is $20.69\text{ \% of }N$ .