Chapter U3.13, Problem 7E

(a)

Interpretation Introduction

Interpretation: The size of the air bag needs to be determined at temperature -5 ^oC and pressure 0.8 atm.

Concept Introduction: The combined gas law is combination of three gas laws that is Boyle’s law, Charles’ law and Gay-Lussac’s law. According to this law the ratio of temperature to the product of pressure and volume is a constant.

Explanation of Solution

The airbag is designed to inflate to 65 L if accident happens at temperature and air pressure 25 ^oC and 1.0 atm respectively. These given values will be initial volume, temperature and pressure.

The new temperature and pressure is -5 ^oC and 0.8 atm respectively.

The temperature in K can be calculated as follows:

  $\begin{array}{l}{T}_1=25{}^{\text{o}}\text{C+273}\text{.15=298}\text{.15 K}\\ T_2=-0.5{}^{\text{o}}\text{C+273}\text{.15=272}\text{.65 K}\end{array}$

The new volume of the air bag can be calculated using the combined gas law as follows:

  $V_2=\frac{P_1{V}_1{T}_2}{P_2{T}_1}$

Here, P₁, V₁, T₁, P₂, V₂ and T₂ is initial pressure, initial volume, initial temperature, final pressure, final volume and final temperature respectively.

Putting the values,

  $\begin{array}{l}{V}_2=\frac{\left(1.0\text{ atm}\right)\left(65\text{ L}\right)\left(272.65\text{ K}\right)}{\left(0.8\text{ atm}\right)\left(298.15\text{ K}\right)}\\ =74.3\text{ L}\end{array}$

Thus, the new volume is 74.3 L.

(b)

Interpretation Introduction

Interpretation: The change in the temperature and air pressure in the location where airbag inflates to 60 L needs to be explained.

Concept Introduction: The combined gas law is combination of three gas laws that is Boyle’s law, Charles’ law and Gay-Lussac’s law. According to this law the ratio of temperature to the product of pressure and volume is a constant.

Explanation of Solution

Now, if volume changes to 60 L, the temperature and air pressure also changes.

According to combine gas law, the relation is as follows:

  $\frac{P_1{V}_1}{T_1}=\frac{P_2{V}_2}{T_2}$

If volume decreases, since pressure and volume are inversely proportional pressure will increase also, temperature tends to decrease as it is directly proportional to the volume.