Question

At a constant pressure what should be the percentage increase in the temperature in Kelvin for a \(10 \%\) increase in volume? (1) \(10 \%\) (2) \(20 \%\) (3) \(5 \%\) (4) \(50 \%\)

Short answer

The percentage increase in temperature is 10%.

Step-by-step solution

Identify the Ideal Gas Law

Recall the ideal gas law: \[ PV = nRT \] Where: \( P \) = Pressure\( V \) = Volume\( n \) = Number of moles\( R \) = Universal gas constant\( T \) = Temperature in Kelvin

Apply constant pressure condition

For a constant pressure, the relationship becomes \[ V \/ T = \text{constant} \]This implies \[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \]

Express volume increase

Given a \(10\%\) increase in volume, let the initial volume be \( V_1 \). Thus, the new volume \( V_2 \) is \[ V_2 = V_1 (1 + 0.10) = 1.10 V_1 \]

Set up the temperature ratio

Using \( \frac{V_1}{T_1} = \frac{1.10 V_1}{T_2} \), we can cancel out \( V_1 \) on both sides:\[ \frac{1}{T_1} = \frac{1.10}{T_2} \]Rearrange to solve for \( T_2 \): \[ T_2 = 1.10 T_1 \]

Calculate the percentage increase

The percentage increase in temperature is given by the formula: \[ \frac{T_2 - T_1}{T_1} \times 100% \]Substitute \( T_2 = 1.10 T_1 \):\[ \frac{1.10 T_1 - T_1}{T_1} \times 100% = \frac{0.10 T_1}{T_1} \times 100% = 10% \]

Conclusion

Therefore, the percentage increase in temperature is \(10\%\).

Key concepts

Temperature-Volume Relationship

The relationship between temperature and volume of an ideal gas is fundamental in understanding many gas behaviors. According to the Ideal Gas Law, expressed as \[ PV = nRT \], when pressure (\(P\)) is held constant, the volume (\(V\)) of a gas is directly proportional to its temperature (\(T\)) in Kelvin. This implies that as the temperature of a gas increases, its volume increases, provided the pressure remains the same.

This relationship can be simplified to \[ \frac{V}{T} = \text{constant} \]. Hence, when the temperature of gas changes from \(T_1\) to \(T_2\), and its volume changes from \(V_1\) to \(V_2\), the ratio is maintained as \[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \]. This equation helps us determine how one variable changes when the other is altered. Understanding this can be particularly useful in predicting the behavior of gases in various environmental and experimental conditions.

Percentage Increase Calculation

Calculating percentage increases is a valuable skill in both chemistry and general mathematics. For scenarios like the given problem, where we need to find the percentage increase in temperature due to a change in volume, we follow a systematic approach.

When volume increases by a certain percentage, say 10%, the new volume \(V_2\) is expressed as \[ V_2 = V_1 \times (1 + \frac{\text{increaseewline\text{percentage}}}{100}) \]. For a 10% increase, this becomes \[ V_2 = V_1 \times 1.10 \].

Next, to find the corresponding change in temperature, we use the ratio \[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \]. Plugging in our volumes, we get \[ \frac{V_1}{T_1} = \frac{1.10 V_1}{T_2} \], which simplifies to \[ T_2 = 1.10 T_1 \]. The percentage increase in temperature can then be calculated as \[ \frac{T_2 - T_1}{T_1} \times 100 \% \]. Substituting \(1.10 T_1\) for \(T_2\): \[ \frac{1.10 T_1 - T_1}{T_1} \times 100 \%= 10 \% \]. This shows that a 10% increase in volume results in a 10% increase in temperature.

Constant Pressure Condition

Understanding the constant pressure condition is crucial when dealing with gas laws. In many practical situations, it's common to keep the pressure constant to isolate and study the behavior of other variables such as temperature and volume.

At constant pressure, the Ideal Gas Law simplifies to \[ V \/ T = \text{constant} \], meaning volume changes directly with temperature. This is aptly described by Charles's Law which states that volume and temperature are directly proportional when pressure is constant.

For example, in the given problem, a constant pressure allowed us to link the percentage increase in volume directly to a percentage increase in temperature. This way, the resulting relationship \[ V_1 \/ T_1 = V_2 \/ T_2 \] can be simplified to relate changes in volume directly to changes in temperature without worrying about pressure variations. By simplifying our equations under constant pressure, we can more easily predict how changes such as heating or cooling will affect the volume a gas occupies.