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Chapter 11: Problem 119

What happens to the volume of a gas in a cylinder with a movable piston if: (a) The pressure is doubled while the temperature is held constant? (b) The temperature is doubled while the pressure is held constant? (c) The pressure and the temperature are both doubled? (The movable piston means the volume of the cylinder, and therefore of the gas, can change.)

Short Answer

Expert verified
(a) When the pressure is doubled while the temperature is held constant, the volume of the gas in the cylinder with a movable piston will be reduced to half of its initial volume: \(V_{new} = \frac{1}{2}V\). (b) When the temperature is doubled while the pressure is held constant, the volume of the gas in the cylinder with a movable piston will be doubled compared to its initial volume: \(V_{new} = 2V\). (c) When both the pressure and the temperature are doubled, the volume of the gas in the cylinder with a movable piston remains the same as its initial volume: \(V_{new} = V\).

Step by step solution

01

(a) Pressure is doubled, temperature constant

We are given that the pressure is doubled (\(P_{new} = 2P\)) while the temperature remains constant (\(T_{new} = T\)). We will use the initial and final conditions in the ideal gas equation to find the effect on volume. Initial Condition: \(PV = nRT\) Final Condition: \((2P)V_{new} = nR(T)\) Now, divide the final condition by the initial condition: \(\frac{(2P)V_{new}}{PV} = \frac{nRT}{nRT}\) Solving for \(V_{new}\): \(V_{new} = \frac{1}{2}V\) Thus, when the pressure is doubled while the temperature is held constant, the volume of the gas in the cylinder with a movable piston will be reduced to half of its initial volume.
02

(b) Temperature is doubled, pressure constant

Now we are given that the temperature is doubled (\(T_{new} = 2T\)) while the pressure remains constant (\(P_{new} = P\)). Again, use initial and final conditions in the ideal gas equation. Initial Condition: \(PV = nRT\) Final Condition: \(P(V_{new}) = nR(2T)\) Now, divide the final condition by the initial condition: \(\frac{P(V_{new})}{PV} = \frac{2nRT}{nRT}\) Solving for \(V_{new}\): \(V_{new} = 2V\) Thus, when the temperature is doubled while the pressure is held constant, the volume of the gas in the cylinder with a movable piston will be doubled compared to its initial volume.
03

(c) Pressure and temperature are doubled

Here, both pressure and temperature are doubled (\(P_{new} = 2P\) and \(T_{new} = 2T\)). Use the initial and final conditions in the ideal gas equation. Initial Condition: \(PV = nRT\) Final Condition: \((2P)V_{new} = nR(2T)\) Now, divide the final condition by the initial condition: \(\frac{(2P)V_{new}}{PV} = \frac{2nRT}{nRT}\) Solving for \(V_{new}\): \(V_{new} = V\) Thus, when both the pressure and the temperature are doubled, the volume of the gas in the cylinder with a movable piston remains the same as its initial volume.

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Most popular questions from this chapter

Regarding temperatures in kelvins: (a) Convert \(-100.5^{\circ} \mathrm{C}\) to kelvins. (b) What is wrong with someone telling you the temperature of something is \(-100.5 \mathrm{~K} ?\) (c) What is the coldest possible temperature in \({ }^{\circ} \mathrm{C} ?\)

If a gas occupies \(2.40 \mathrm{~L}\) at \(1 \mathrm{~atm}\) and \(22^{\circ} \mathrm{C}\), at what Celsius temperature will it occupy \(7.20 \mathrm{~L}\) at \(1 \mathrm{~atm} ?\)

"The older I get, the fewer hairs I have on my head." What kind of relationship (proportion or inverse proportion) exists between this gentleman's age and his hair? Explain your answer.

Rewrite the ideal gas law solving for \(n\). Also show how all units cancel to leave you with just units of moles.

What do we mean by inverse proportionality? By direct proportionality? Give an example of each using the way the pressure of a gas depends on something else.
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