Q5.33P - Chemistry: Molecular Nature Of Matter And Change

Question

A gas-filled weather balloon with a volume of 65.0 L is released at sea-level conditions of 745 torr and $\mathbf{25}\mathbf{°}\mathbf{C}$ . The balloon can expand to a maximum volume of 835 L. When the balloon rises to an altitude at which the temperature is $\mathbf{-}\mathbf{5}\mathbf{°}\mathbf{C}$ and the pressure is 0.066 atm, will it reach its maximum volume?

Step-by-Step Solution

Answer

Yes, the gas filled balloon will reach its maximum volume in the given conditions.

Step 1: Ideal gas law

The ideal gas law equation is,

 $\mathbf{PV}\mathbf{=}\mathbf{nRT}$

 

 

The ideal gas law for the given condition can be written as,

 

 $\frac{{\mathbf{P}}_{\mathbf{1}}{\mathbf{V}}_{\mathbf{1}}}{{\mathbf{T}}_{\mathbf{1}}}\mathbf{=}\frac{{\mathbf{P}}_{\mathbf{2}}{\mathbf{V}}_{\mathbf{2}}}{{\mathbf{T}}_{\mathbf{2}}}$

 

(Moles of the sample are fixed)

 

Here, 

 

 ${\text{V}}_{\text{1}}$ is the initial volume (65.0 L).

${\mathrm{V}}_2$  is the final volume.

 data-custom-editor="chemistry" ${\mathrm{T}}_1$ is the initial temperature ( data-custom-editor="chemistry" $25°\mathrm{C}$ ).

data-custom-editor="chemistry" ${\mathrm{T}}_2$  is the final temperature ( data-custom-editor="chemistry" $-5°\mathrm{C}$).

data-custom-editor="chemistry" ${\mathrm{P}}_1$  is the initial temperature (745 torr).

data-custom-editor="chemistry" ${\mathrm{P}}_2$  is the final pressure (0.066 atm).

Step 2: Determination of the final volume.

Before putting the values, we need to convert the given units in desired units,

$\begin{array}{c}{\mathbf{T}}_{\mathbf{1}}\mathbf{=}\mathbf{25}\mathbf{°}\mathbf{C}\mathbf{=}\mathbf{25}\mathbf{+}\mathbf{273}\mathbf{=}\mathbf{298}\mathbf{K}\\ {\mathbf{T}}_{\mathbf{2}}\mathbf{=}\mathbf{-}\mathbf{5}\mathbf{°}\mathbf{C}\mathbf{=}\mathbf{-}\mathbf{5}\mathbf{+}\mathbf{273}\mathbf{=}\mathbf{268}\mathbf{K}\end{array}$

And,

 $\begin{array}{c}\mathbf{1}\mathbf{\text{\hspace{0.17em}\hspace{0.17em}}}\mathbf{torr}\mathbf{=}\mathbf{0}\mathbf{.00132}\mathbf{atm}\\ \mathbf{745}\mathbf{torr}\mathbf{=}\mathbf{745}\mathbf{torr}\mathbf{\times }\frac{\mathbf{0}\mathbf{.}\mathbf{00132}\mathbf{atm}}{\mathbf{1}\mathbf{torr}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{983}\mathbf{atm}\end{array}$

 

 

The final volume of the gas is,

$\frac{{\mathrm{P}}_1{\mathrm{V}}_1}{{\mathrm{T}}_1}=\frac{{\mathrm{P}}_2{\mathrm{V}}_2}{{\mathrm{T}}_2}$

Hence,

 $\begin{array}{c}{\mathrm{V}}_2=\frac{{\mathrm{P}}_1{\mathrm{V}}_1}{{\mathrm{T}}_1}\times \frac{{\mathrm{T}}_2}{{\mathrm{P}}_2}\\ {\mathrm{V}}_2=\frac{0.983\mathrm{atm}\times 65.0\mathrm{L}}{298\mathrm{K}}\times \frac{268\mathrm{K}}{0.066\mathrm{atm}}\\ {\mathrm{V}}_2=870.6\mathrm{L}\end{array}$

 

 

Thus, the final volume of the gas in the balloon is 870.6 L which is more than its maximum volume of 835 L.

 

And the gas filled balloon will reach its maximum volume in these conditions.