Problem 84
Question
The molecules of a given mass of a gas have RMS speed of \(200 \mathrm{~m} / \mathrm{s}\) at \(300 \mathrm{~K}\) and \(1,00,000\) bar pressure. When the absolute temperature is doubled and the pressure is halved, the RMS speed of molecules will become (a) \(200 \mathrm{~m} / \mathrm{s}\) (b) \(400 \mathrm{~m} / \mathrm{s}\) (c) \(100 \mathrm{~m} / \mathrm{s}\) (d) \(200 \sqrt{2} \mathrm{~m} / \mathrm{s}\)
Step-by-Step Solution
Verified Answer
The RMS speed of the molecules will become (d) \(200\sqrt{2}\) m/s after the changes in temperature and pressure.
1Step 1 - Review Concepts
Understand that the root-mean-square (RMS) speed of the molecules of a gas is related to its temperature and pressure. The RMS speed, denoted by 'v', is given by the formula derived from the kinetic theory of gases: \(v = \sqrt{\frac{3kT}{m}}\), where 'k' is the Boltzmann constant, 'T' is the absolute temperature, and 'm' is the mass of a single molecule of the gas. When dealing with a fixed amount of gas, the RMS speed is directly proportional to the square root of the absolute temperature, and it is independent of pressure, given the mass stays constant.
2Step 2 - Apply the Changes in Temperature
According to the given information, when the absolute temperature is doubled, the new temperature becomes \(T' = 2 \cdot 300 \ K = 600 \ K\). Since we are dealing with a given mass of gas, the change in RMS speed resulting from the change in temperature can be expressed as \(v' = v \sqrt{\frac{T'}{T}}\).
3Step 3 - Compute the New RMS Speed
Substitute the known values into the new RMS speed formula to calculate the new speed: \(v' = 200 \: m/s \cdot \sqrt{\frac{600 \: K}{300 \: K}} = 200 \: m/s \cdot \sqrt{2} = 200\sqrt{2} \: m/s\).
4Step 4 - Address the Change in Pressure
Since the RMS speed of gas molecules is not dependent on pressure for a given mass and molecule, halving the pressure does not affect the RMS speed. The final RMS speed remains the same as calculated after adjusting for the change in temperature.
Key Concepts
Kinetic Theory of GasesBoltzmann ConstantAbsolute Temperature in Gas Behavior
Kinetic Theory of Gases
The kinetic theory of gases is a fundamental principle that helps us understand the behavior of gases at the molecular level. This theory is grounded on the assumption that gases consist of many small particles which are in constant, random motion.
According to this theory, the pressure exerted by a gas is a result of collisions between the gas molecules and the walls of their container. The temperature of the gas reflects the average kinetic energy of these molecules. Therefore, when we say that a gas is at a higher temperature, it implies that, on average, its molecules are moving faster. The RMS (root-mean-square) speed is especially important because it relates to the speed at which these molecules travel. It is the square root of the average of the squares of the velocities of each molecule, providing an effective measure of their speed.
In our case study, the RMS speed was initially given, along with the gas's temperature and pressure. To determine how changes in conditions affect the RMS speed, the kinetic theory of gases provides the necessary concepts, as it suggests that such speed depends on the temperature and is independent of pressure.
According to this theory, the pressure exerted by a gas is a result of collisions between the gas molecules and the walls of their container. The temperature of the gas reflects the average kinetic energy of these molecules. Therefore, when we say that a gas is at a higher temperature, it implies that, on average, its molecules are moving faster. The RMS (root-mean-square) speed is especially important because it relates to the speed at which these molecules travel. It is the square root of the average of the squares of the velocities of each molecule, providing an effective measure of their speed.
In our case study, the RMS speed was initially given, along with the gas's temperature and pressure. To determine how changes in conditions affect the RMS speed, the kinetic theory of gases provides the necessary concepts, as it suggests that such speed depends on the temperature and is independent of pressure.
Boltzmann Constant
The Boltzmann constant, symbolized as 'k', is a crucial bridge connecting the macroscopic and microscopic worlds, linking variables that describe gases at a scale perceptible to humans, like temperature, to the energy levels observable only at the molecular or atomic scale.
This constant appears in many fundamental equations in physics and chemistry, including the one for RMS speed of gas molecules: \( v = \sqrt{\frac{3kT}{m}} \). Here, 'T' stands for absolute temperature, and 'm' is the mass of an individual gas molecule. With the Boltzmann constant playing a role in this relationship, it enables us to calculate the RMS speed, providing us a way to translate the average kinetic energy of gas molecules into a temperature we can measure.
Understanding the role of the Boltzmann constant is vital as it allows us to make predictions about the behavior of gas molecules. For example, if we know the absolute temperature and the mass of the gas molecules, we can estimate their average speed.
This constant appears in many fundamental equations in physics and chemistry, including the one for RMS speed of gas molecules: \( v = \sqrt{\frac{3kT}{m}} \). Here, 'T' stands for absolute temperature, and 'm' is the mass of an individual gas molecule. With the Boltzmann constant playing a role in this relationship, it enables us to calculate the RMS speed, providing us a way to translate the average kinetic energy of gas molecules into a temperature we can measure.
Understanding the role of the Boltzmann constant is vital as it allows us to make predictions about the behavior of gas molecules. For example, if we know the absolute temperature and the mass of the gas molecules, we can estimate their average speed.
Absolute Temperature in Gas Behavior
Absolute temperature, usually measured in kelvins (K), is a scale that relates directly to the energy contained within particles. It’s absolute because it starts from absolute zero, where theoretically, molecular motion ceases.
In relation to gas behavior, temperature plays a key role; it determines the kinetic energy of the gas molecules, which in turn affects their movement speed. The RMS speed of gas molecules increases with the square root of absolute temperature, illustrating that a higher temperature means faster moving molecules.
The exercise highlighted a scenario where the gas temperature is doubled, leading to an increase in RMS speed by a factor of the square root of two. It's essential to note that this increase in speed is driven solely by the temperature change and not influenced by the halving of the pressure, affirming that in kinetic theory, the RMS speed depends on temperature and not pressure for a given mass of gas.
In relation to gas behavior, temperature plays a key role; it determines the kinetic energy of the gas molecules, which in turn affects their movement speed. The RMS speed of gas molecules increases with the square root of absolute temperature, illustrating that a higher temperature means faster moving molecules.
The exercise highlighted a scenario where the gas temperature is doubled, leading to an increase in RMS speed by a factor of the square root of two. It's essential to note that this increase in speed is driven solely by the temperature change and not influenced by the halving of the pressure, affirming that in kinetic theory, the RMS speed depends on temperature and not pressure for a given mass of gas.
Other exercises in this chapter
Problem 81
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At moderate pressure, the compressibility factor for a gas is given as: \(Z=1+0.35 P\) \(-\frac{168}{T} \cdot P\), where \(P\) is in bar and \(T\) is in Kelvin.
View solution Problem 85
At STP, the order of the RMS speed of molecules of \(\mathrm{H}_{2}, \mathrm{~N}_{2}, \mathrm{O}_{2}\) and HBr gases is (a) \(\mathrm{H}_{2}>\mathrm{N}_{2}>\mat
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