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Chapter 5: Problem 22

A sample of air occupies \(3.8 \mathrm{~L}\) when the pressure is \(1.2 \mathrm{~atm} .\) (a) What volume does it occupy at \(6.6 \mathrm{~atm} ?\) (b) What pressure is required in order to compress it to \(0.075 \mathrm{~L} ?\) (The temperature is kept constant.)

Short Answer

Expert verified
For part (a), the final volume is approximately 0.69 L, and for part (b), the final pressure is approximately 60.8 atm.

Step by step solution

01

Applying Ideal Gas Law - Part (a)

For part (a), the relationship between the initial and final states of pressure and volume can be expressed using the ideal gas law. Thus, the equation is \(P_1V_1 = P_2V_2\). Plugging in the given initial pressure \(P_1 = 1.2 \, \mathrm{atm}\) and volume \(V_1 = 3.8 \, \mathrm{L}\), as well as the final pressure \(P_2 = 6.6 \, \mathrm{atm}\), we can solve for the final volume \(V_2\). That gives us \(V_2 = \frac{P_1V_1}{P_2}\).
02

Calculation of Final Volume - Part (a)

Substituting the given values into the equation from Step 1, we find \(V_2 = \frac{(1.2 \, \mathrm{atm})(3.8 \, \mathrm{L})}{6.6 \, \mathrm{atm}}\). This results in \(V_2 \approx 0.69 \, \mathrm{L}\). This is the volume the air would occupy at a pressure of 6.6 atm.
03

Applying Ideal Gas Law - Part (b)

For part (b), the same ideal gas law translates to \(P_1V_1 = P_2V_2\). Given the initial pressure \(P_1 = 1.2 \, \mathrm{atm}\) and volume \(V_1 = 3.8 \, \mathrm{L}\), as well as the final volume \(V_2 = 0.075 \, \mathrm{L}\), we can find the final pressure \(P_2\). Therefore \(P_2 = \frac{P_1V_1}{V_2}\).
04

Calculation of Final Pressure - Part (b)

Substituting the given values into the equation from Step 3, we find \(P_2 = \frac{(1.2 \, \mathrm{atm})(3.8 \, \mathrm{L})}{0.075 \, \mathrm{L}}\). This results in \(P_2 \approx 60.8 \, \mathrm{atm}\). This is the pressure required in order to compress the air to a volume of 0.075 L.

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