Problem 5 - Chemistry The Central Science In Si Units

Question

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?

Step-by-Step Solution

Answer

The temperature of the gas doubles.

Step 1: Understand the Relationship

According to the ideal gas law, given by \( PV = nRT \), for a constant volume, the relationship between pressure and temperature is defined by \( P \propto T \). This means that pressure is directly proportional to temperature when the volume and amount of gas are constant.

Step 2: Analyze the Change in Pressure

In the problem, we are told that the pressure of the gas is doubled. This means that the new pressure \( P_2 \) is twice the initial pressure \( P_1 \). Mathematically, this is expressed as \( P_2 = 2P_1 \).

Step 3: Relate Pressure Change to Temperature Change

Using the direct relationship \( P \propto T \), if the pressure is doubled, then the temperature must also double to maintain the equation because \( \frac{P_2}{P_1} = \frac{T_2}{T_1} \). So, if \( P_2 = 2P_1 \), then \( T_2 = 2T_1 \).

Step 4: Conclusion

The temperature of the gas must also double for the given condition, maintaining the direct proportionality in the ideal gas law equation as both are directly linked when the volume is constant.

Key Concepts

Pressure-Temperature Relationship

The relationship between pressure and temperature in an ideal gas is a fundamental concept in physics. According to the ideal gas law, expressed as \( PV = nRT \), where:

• \( P \) stands for pressure
• \( V \) represents volume
• \( n \) is the number of moles of the gas
• \( R \) is the ideal gas constant
• \( T \) denotes temperature

When we hold volume constant, the law simplifies. The pressure \( P \) of the gas is directly related to its temperature \( T \). This is expressed as \( P \propto T \).
In simple terms, if you change the temperature of the gas, its pressure will alter in the same proportion, assuming the volume doesn’t change. For instance, doubling the pressure of the gas will result in a doubling of temperature. This clear association helps predict how a gas will behave under different conditions of temperature and pressure.

Constant Volume

The scenario of a constant volume is crucial in understanding the behavior of gases. In this context, the container holding the gas does not change size, meaning the space available for the gas's particles to move remains fixed.
This concept is significant because it isolates the variables of pressure and temperature. When studying the ideal gas law, maintaining a constant volume allows us to closely examine how changes in temperature and pressure influence each other without being affected by changes in the container's size.

• This simplifies the analysis to focus on just the relationship between pressure and temperature.
• Any change in thermal energy (temperature) directly impacts the pressure exerted by the gas.
• No other gas properties (like volume alterations) complicate the problem.

Understanding the implications of constant volume simplifies the conditions purported in theoretical and practical situations, such as when heating a gas in a closed system.

Direct Proportionality

Direct proportionality is a term that describes a relationship where two quantities increase or decrease in unison. In the context of the ideal gas law, pressure and temperature are directly proportional at a constant volume.
This means if pressure increases, temperature does as well, and vice versa. If one quantity were to double, the other would follow suit. Mathematically, if we express this proportional relationship as \( P \propto T \), any alteration in pressure leads to an equivalent change in temperature.

• This is calculated using the formula \( \frac{P_2}{P_1} = \frac{T_2}{T_1} \).
• Such relationships help scientists predict and control the behavior of gases in different conditions.
• Practical applications include everything from calculating outcomes in chemical reactions to designing engines.

Understanding direct proportionality helps to logically reason why physical phenomena occur. By knowing if one variable changes, how the other will react can easily be predicted, allowing for accurate forecasting of results in experimental and theoretical settings.