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Chapter 5: Problem 20

What is the effect of the following on the volume of \(1 \mathrm{~mol}\) of an ideal gas? (a) The pressure is tripled (at constant \(T\) ). (b) The absolute temperature is increased by a factor of 3.0 (at constant \(P\) ). (c) Three more moles of the gas is added (at constant \(P\) and \(T\) ).

Short Answer

Expert verified
(a) Volume is reduced to one-third. (b) Volume is tripled. (c) Volume is quadrupled.

Step by step solution

01

Understanding the Ideal Gas Law

The ideal gas law is given by the equation \( PV = nRT \), where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is the temperature in Kelvin. This equation is crucial for determining the effect of changes in pressure, temperature, and the number of moles on the volume of the gas.
02

Effect of Tripling the Pressure (at constant T)

According to the ideal gas law, if the pressure \( P \) is tripled while the temperature \( T \) remains constant, the volume \( V \) will be inversely proportional to the pressure. Mathematically, this can be expressed as: \( P_1V_1 = P_2V_2 \). When \( P_2 = 3P_1 \), it follows that \( V_2 = \frac{V_1}{3} \). Therefore, the volume is reduced to one-third of its original volume.
03

Effect of Tripling the Absolute Temperature (at constant P)

If the absolute temperature \( T \) is increased by a factor of 3 while the pressure \( P \) remains constant, the volume \( V \) will be directly proportional to the temperature. Using the relationship \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \), and substituting \( T_2 = 3T_1 \), it follows that \( V_2 = 3V_1 \). Thus, the volume will be tripled.
04

Effect of Adding Three More Moles of the Gas (at constant P and T)

When three more moles are added to the initial one mole (at constant pressure and temperature), the total number of moles becomes \( n_2 = 4n_1 \). The volume \( V \) is directly proportional to the number of moles, as given by \( \frac{V_2}{V_1} = \frac{n_2}{n_1} \). Therefore, \( V_2 = 4V_1 \). Hence, the volume will be quadrupled.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Effect of pressure on gas volume
The relationship between pressure and volume in an ideal gas is inverse, as described by the Ideal Gas Law: \(PV = nRT\). When pressure increases, volume decreases, and vice versa. Imagine you're squeezing a balloon. When you apply more pressure, the balloon shrinks. This is because the gas molecules have less space to move around.

For instance, if we triple the pressure on a gas (keeping temperature constant), the new volume will be a third of what it was initially. Mathematically, when \(P_2 = 3P_1\), it leads to \(V_2 = \frac{V_1}{3}\). This means that tripling the pressure reduces the volume to one-third.
Effect of temperature on gas volume
Temperature and volume in an ideal gas are directly proportional, which is a part of the Ideal Gas Law. When we heat a gas, its molecules move faster, causing the gas to expand.

Consider a scenario where we increase the temperature of an ideal gas by three times while keeping the pressure constant. According to the law, the volume will also triple. This relationship is depicted with the equation \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \). If \(T_2 = 3T_1\), then \(V_2 = 3V_1\). Therefore, tripling the temperature results in tripling the gas volume.
Effect of moles on gas volume
The number of moles of a gas is directly proportional to its volume, assuming pressure and temperature are constant. More moles mean more gas molecules, leading to a larger volume to accommodate these molecules.

If we add three more moles to an existing one mole of gas, the total becomes four moles. Hence, the volume increases fourfold. This relationship can be represented by the equation \( \frac{V_2}{V_1} = \frac{n_2}{n_1} \). Therefore, with \( n_2 = 4n_1\), we have \(V_2 = 4V_1\). Adding more moles significantly increases the gas volume.

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