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\(\bullet$$\bullet\) We have two equal-size boxes, \(\mathrm{A}\) and \(\mathrm{B}\) . Each box contains gas that behaves as an ideal gas. We insert a thermometer into each box and find that the gas in box \(A\) is at a temperature of \(50^{\circ} \mathrm{C}\) while the gas in box \(B\) is at \(10^{\circ} \mathrm{C} .\) This is all we know about the gas in the boxes. Which of the following statements must be true? Which could be true? (a) The pressure in \(A\) is higher than in \(B\) . (b) There are more molecules in \(A\) than in \(B\) . (c) \(A\) and \(B\) cannot contain the same type of gas. (d) The molecules in \(A\) have more average kinetic energy per molecule than those in \(B\) (e) The molecules in \(A\) are moving faster than those in \(B\) . Explain the reasoning behind your answers.

Step by step solution

01

Understanding Temperature and Pressure

The relationship between temperature and pressure for an ideal gas in a fixed volume can be described by the equation \( P \propto T \), meaning pressure \( P \) is directly proportional to temperature \( T \) in Kelvin. Since box A has a higher temperature than box B, the pressure in A must be higher than in B.

02

Calculating Molecules: Number Density Remains Unchanged

The number of molecules in a gas at constant volume and temperature is given by the ideal gas law, and since the volume is constant in both boxes, a temperature difference does not necessarily affect the number of molecules. Without more specific details, we cannot definitively say that there are more molecules in A than in B.

03

Determining Type of Gas With Temperature

From the information given, both boxes can indeed contain the same type of ideal gas despite the difference in temperature. The variation in temperature alone does not inherently require different gases to be present.

04

Average Kinetic Energy: Temperature Dependent

The average kinetic energy per molecule of a gas is directly proportional to its temperature in Kelvin: \( KE \propto T \). Therefore, since the temperature of box A is higher than that of box B, the average kinetic energy of the molecules in A must be greater than in B.

05

Molecular Speed and Temperature

The speed of gas molecules is also related to their temperature, as described by the equation \( v \propto \sqrt{T} \). Thus, because box A has a higher temperature than box B, the molecules in A must be moving faster than those in B.

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Key Concepts

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

Temperature and Pressure Relationship

In the world of ideal gases, understanding the link between temperature and pressure is crucial for grasping how gases behave in different conditions. For any ideal gas, at a constant volume, the pressure (\( P \)) is directly proportional to its absolute temperature (\( T \)). This is expressed by the equation: \( P \propto T \). This relationship illustrates that as the temperature of a gas increases, so does its pressure, provided the volume doesn't change.

Applying this concept to our exercise, the gas in box A, with a higher temperature of \(50^{\circ} \text{C} \), will exert more pressure than the gas in box B at \(10^{\circ} \text{C} \). This outcome is inevitable even though we do not know the exact amount or type of gas in the boxes, thanks to the ideal gas behavior.

• The pressure is affected by changes in temperature.
• The relationship holds as long as the volume stays the same.
• Knowing one variable's change allows predictions about the other's change.

Kinetic Energy and Temperature

Kinetic energy is a measure of an object's motion energy, and for gas molecules, it is closely linked to temperature. The average kinetic energy of gas particles is directly proportional to the temperature in Kelvin: \( KE \propto T \). This means warmer gases have more energetic molecules than cooler ones.

In our scenario, the molecules in box A, being at a higher temperature, have greater kinetic energy than those in box B. Thus, option (d) in the exercise must be true as it directly correlates to the well-established principles of thermodynamics.

• Temperature changes directly affect kinetic energy.
• Lower temperature corresponds to lower kinetic energy.
• Higher temperature means there's more motion energy in the molecules.

Molecular Speed and Temperature

The speed of molecules in a gas is not only a key aspect of understanding gas behavior but also related to kinetic theories and thermodynamics. Typically, molecular speed (\( v \)) is proportional to the square root of the temperature, as expressed in the equation: \( v \propto \sqrt{T} \).

A rise in temperature increases molecules' speed. Consequently, the warmer gas in box A will have faster-moving molecules compared to those in cooler box B. This aligns with the observation that options (d) and (e) in the exercise are interrelated—greater temperature translates to both higher kinetic energy and faster molecular speed.

• Increased temperature implies higher energy per molecule.
• Faster molecule speeds result from increased temperatures.
• The relationship is essential to predict gas behavior during temperature changes.