Chapter U3, Problem 8RE

Interpretation Introduction

Interpretation: The new volume of the balloon needs to be determined.

Concept Introduction:

The ideal gas equation is represented as follows:

  $PV=nRT$

Here, P is pressure, V is volume, n is number of moles, R is Universal gas constant and T is temperature of the gas.

Answer to Problem 8RE

90.72 L.

Explanation of Solution

Given Information: The initial volume, temperature and pressure of the air in balloon is 75 L, 28 ^oC and 1 atm respectively. The final pressure and temperature is 0.8 atm and 18 ^oC respectively.

Calculation:

First convert the given temperature to K as follows:

  $0{}^{\text{o}}\text{C}=\text{273}\text{.15 K}$

Thus,

  $28{}^{\text{o}}\text{C}=28+\text{273}\text{.15 K}=\text{301}\text{.15 K}$

Thus,

  $T_1=301.15\text{ K}$

Also,

  $18{}^{\text{o}}\text{C}=18+\text{273}\text{.15 K}=\text{291}\text{.15 K}$

Thus,

  $T_2=291.15\text{ K}$

The ideal gas equation is as follows:

  $PV=nRT$

On rearranging, the number of moles of the air in the balloon can be calculated as follows:

  $n=\frac{P_1{V}_1}{R{T}_1}$

Putting the values,

  $n=\frac{\left(1.0\text{ atm}\right)\left(75.0\text{ L}\right)}{\left(0.082\text{ L atm/K mol}\right)\left(301.15\text{ K}\right)}=3.04\text{ mol}$

The number of moles of air in the balloon remains the same even at the top of the mountain thus; new volume of the balloon can be calculated as follows:

  $V_2=\frac{nR{T}_2}{P_2}$

Putting the values,

  $\begin{array}{l}{V}_2=\frac{\left(3.04\text{ mol}\right)\left(0.082\text{ L atm/K mol}\right)\left(291.15K\right)}{\left(0.8\text{ atm}\right)}\\ =90.72\text{ L}\end{array}$

Conclusion

Thus, the new volume of the air in balloon is 90.72 L.