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Chapter 10: Problem 5

Suppose you have a fixed amount of an ideal gas at a constant volume. If the pressure of the gas is doubled while the volume is held constant, what happens to its temperature? [Section 10.4]

Short Answer

Expert verified
The temperature of an ideal gas remains the same when its pressure is doubled while its volume is held constant, as T2 = T1.

Step by step solution

01

Write down the initial and final conditions of the gas

We are given that the pressure of an ideal gas doubles while its volume is held constant. Let's denote the initial pressure as P1 and the final pressure as P2. Since the pressure doubles, P2 = 2 * P1. Let's denote the initial temperature as T1 and the final temperature as T2, which is what we need to find.
02

Apply the ideal gas law to the initial and final conditions

According to the ideal gas law, PV = nRT. For the initial condition, we have P1 * V = n * R * T1. For the final condition, we have P2 * V = n * R * T2. Since the volume, number of moles, and gas constant are the same for both conditions, we can write V = V, n = n, and R = R.
03

Solve for the final temperature in terms of the initial temperature

Let's now express the ideal gas law for the final condition in terms of the initial condition. We have (2 * P1) * V = n * R * T2. Now we can substitute P1 * V = n * R * T1 into this equation: (2 * P1) * V = 2 * (n * R * T1) => 2 * (P1 * V) = 2 * (n * R * T1). Dividing both sides by 2, we get: P1 * V = n * R * T1. Now, we can solve for the final temperature T2: T2 = T1.
04

Conclude the change in temperature

The final temperature T2 is equal to the initial temperature T1 after the pressure of an ideal gas is doubled while its volume is held constant. This means that when the pressure of an ideal gas doubles at constant volume, the temperature of the gas remains the same.

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Most popular questions from this chapter

A 6.53-g sample of a mixture of magnesium carbonate and calcium carbonate is treated with excess hydrochloric acid. The resulting reaction produces \(1.72 \mathrm{~L}\) of carbon dioxide gas at \(28^{\circ} \mathrm{C}\) and 743 torr pressure. (a) Write balanced chemical equations for the reactions that occur between hydrochloric acid and each component of the mixture. (b) Calculate the total number of moles of carbon dioxide that forms from these reactions. (c) Assuming that the reactions are complete, calculate the percentage by mass of magnesium carbonate in the mixture.

(a) Amonton's law expresses the relationship between pressure and temperature. Use Charles's law and Boyle's law to derive the proportionality relationship between \(P\) and \(T\). (b) If a car tire is filled to a pressure of \(32.0 \mathrm{lbs} / \mathrm{in}^{2}\) (psi) measured at \(75^{\circ} \mathrm{F}\), what will be the tire pressure if the tires heat up to \(120^{\circ} \mathrm{F}\) during driving?

A quantity of \(\mathrm{N}_{2}\) gas originally held at \(5.25 \mathrm{~atm}\) pressure in a \(1.00\) - \(\mathrm{L}\) container at \(26^{\circ} \mathrm{C}\) is transferred to a \(12.5\) - \(\mathrm{L}\) container at \(20^{\circ} \mathrm{C}\). A quantity of \(\mathrm{O}_{2}\) gas originally at \(5.25\) atm and \(26^{\circ} \mathrm{C}\) in a \(5.00\)-L container is transferred to this same container. What is the total pressure in the new container?

A neon sign is made of glass tubing whose inside diameter is \(2.5 \mathrm{~cm}\) and whose length is \(5.5 \mathrm{~m}\). If the sign contains neon at a pressure of \(1.78\) torr at \(35^{\circ} \mathrm{C}\), how many grams of neon are in the sign? (The volume of a cylinder is \(\pi r^{2} h\).)

Carbon dioxide, which is recognized as the major contributor to global warming as a "greenhouse gas," is formed when fossil fuels are combusted, as in electrical power plants fueled by coal, oil, or natural gas. One potential way to reduce the amount of \(\mathrm{CO}_{2}\) added to the atmosphere is to store it as a compressed gas in underground formations. Consider a 1000 -megawatt coal-fired power plant that produces about \(6 \times 10^{6}\) tons of \(\mathrm{CO}_{2}\) per year. (a) Assuming ideal-gas behavior, \(1.00 \mathrm{~atm}\), and \(27^{\circ} \mathrm{C}\), calculate the volume of \(\mathrm{CO}_{2}\) produced by this power plant. (b) If the \(\mathrm{CO}_{2}\) is stored underground as a liquid at \(10^{\circ} \mathrm{C}\) and \(120 \mathrm{~atm}\) and a density of \(1.2 \mathrm{~g} / \mathrm{cm}^{3}\), what volume does it possess? (c) If it is stored underground as a gas at \(36^{\circ} \mathrm{C}\) and \(90 \mathrm{~atm}\), what volume does it occupy?
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