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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
When the pressure of an ideal gas is doubled at constant volume and fixed amount, its temperature also doubles. This can be derived from the modified Ideal Gas Law Equation \(P_1/T_1 = P_2/T_2\), which leads to the final temperature \(T_2 = 2T_1\).

Step by step solution

01

Ideal Gas Law Formula

The ideal gas law formula is given by: \(PV = nRT\), where: - P is the pressure in pascals (Pa), - V is the volume in cubic meters (m³), - n is the amount of gas in moles, - R is the gas constant (\(8.31 J/(mol·K)\)), - T is the temperature in kelvin (K). Since the volume V and the amount of the gas n are constant, we can write the equation as: \(P_1/T_1 = P_2/T_2\), where \(P_1\) and \(P_2\) are the initial and final pressures and \(T_1\) and \(T_2\) are the initial and final temperatures.
02

Given data

We are given: \(- P_1 = \) initial pressure \(- P_2 = 2P_1\) (pressure is doubled) \(- V = \) constant volume \(- n = \) fixed amount of gas We are supposed to find how the temperature changes when the pressure is doubled keeping the volume constant.
03

Using the modified Ideal Gas Law Equation

As we have already derived a modified equation for our specific conditions: \(P_1/T_1 = P_2/T_2\) Now we need to substitute the given data into the equation: \(P_1/T_1 = (2P_1)/T_2\)
04

Solving for T2

Now, we'll solve for \(T_2\) (the final temperature), which is what we are supposed to find out: \(\frac{P_1}{T_1} = \frac{2P_1}{T_2}\) \(\Rightarrow T_2 = 2T_1\) We find that the final temperature \(T_2\) is double the initial temperature \(T_1\). Thus, when the pressure of an ideal gas is doubled at constant volume and fixed amount, its temperature also doubles.

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Most popular questions from this chapter

Torricelli, who invented the barometer, used mercury in its construction because mercury has a very high density, which makes it possible to make a more compact barometer than one based on a less dense fluid. Calculate the density of mercury using the observation that the column of mercury is $760 \mathrm{~mm}\( high when the atmospheric pressure is \)1.01 \times 10^{5} \mathrm{~Pa}$. Assume the tube containing the mercury is a cylinder with a constant cross-sectional area.

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, $101.3 \mathrm{kPa}\(, and \)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 $12.16 \mathrm{MPa}\( and a density of \)1.2 \mathrm{~g} / \mathrm{cm}^{3},$ what volume does it possess? (c) If it is stored underground as a gas at \(30^{\circ} \mathrm{C}\) and \(7.09 \mathrm{MPa}\), what volume does it occupy?

In the Dumas-bulb technique for determining the molar mass of an unknown liquid, you vaporize the sample of a liquid that boils below $100^{\circ} \mathrm{C}$ in a boiling-water bath and determine the mass of vapor required to fill the bulb. From the following data, calculate the molar mass of the unknown liquid: mass of unknown vapor, \(1.012 \mathrm{~g}\); volume of bulb, \(354 \mathrm{~cm}^{3}\); pressure, \(98.93 \mathrm{kPa} ;\) temperature, $99{ }^{\circ} \mathrm{C}$.

A set of bookshelves rests on a hard floor surface on four legs, each having a cross-sectional dimension of \(4.0 \times 5.0 \mathrm{~cm}\) in contact with the floor. The total mass of the shelves plus the books stacked on them is $200 \mathrm{~kg}$. Calculate the pressure in atmospheres exerted by the shelf footings on the surface.

In the United States, barometric pressures are generally reported in inches of mercury (in. Hg). On a beautiful summer day in Chicago, the barometric pressure is 30.45 in. \(\mathrm{Hg}\). (a) Convert this pressure to torr. (b) Convert this pressure to atm.
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