Question

It is desired to increase the volume of 800 cc of a gas by \(20 \%\) keeping the pressure constant. To what temperature the gas be heated, if the initial temperature is \(22^{\circ} \mathrm{C}\) ? (1) \(360^{\circ} \mathrm{C}\) (2) \(87 \mathrm{~K}\) (3) \(454 \mathrm{~K}\) (4) \(81^{\circ} \mathrm{C}\)

Short answer

The gas must be heated to \(81^{\circ} \mathrm{C} \).

Step-by-step solution

Identify the Given Variables

Given: Initial volume of the gas, \(V1 = 800 \ \mathrm{cc} \), initial temperature, \(T1 = 22^{\circ} \mathrm{C} \,(or \, 295\ \mathrm{K}) \). We want to increase the volume by \(20\%\).

Calculate the Final Volume

The final volume, \(V2 \), after a \(20\%\) increase is calculated as: \[ V2 = V1 + \frac{20}{100} \cdot V1 = 800 + \frac{20}{100} \cdot 800 = 960 \ \mathrm{cc} \]

Apply the Ideal Gas Law

Since the pressure is constant, use Charles's Law: \[ \frac{V1}{T1}=\frac{V2}{T2} \] Substitute the known values: \[ \frac{800}{295} = \frac{960}{T2} \]

Solve for the Final Temperature

Rearrange the equation to solve for \(T2 \): \[ T2 = \frac{960 \cdot 295}{800} = 354 \mathrm{K} \]

Convert to Celsius

Convert the temperature from Kelvin to Celsius: \[ T2 = 354 \mathrm{K} - 273 = 81^{\circ} \mathrm{C} \]

Key concepts

Charles's Law

Charles's Law is a fundamental principle in thermodynamics. It describes how gases tend to expand when heated. This law states that the volume of an ideal gas is directly proportional to its absolute temperature, provided the pressure remains constant. Mathematically, it is expressed as \(\frac{V1}{T1} = \frac{V2}{T2}\). Here, \(V1\) and \(T1\) represent the initial volume and temperature, while \(V2\) and \(T2\) represent the final volume and temperature. This relationship helps us understand and predict how a gas will respond to changes in temperature if the pressure doesn't change.
In our problem, we used Charles's Law to determine the new temperature required to increase the gas volume by 20%. When applying Charles's Law, be sure to always use the absolute temperature (measured in Kelvin) for calculations.

Gas Volume Expansion

Gas volume expansion occurs when the temperature of a gas increases, causing its molecules to move more vigorously. This increased motion leads to the molecules taking up more space. In simpler terms, when you heat a gas, it expands if the pressure is kept constant.
Consider our example: the volume of gas increases from 800 cc to 960 cc when heated. Here’s how we calculated it:

• We started with an initial volume, \(V1 = 800 \text{cc}\).
• We wanted to increase this by 20%, so we computed the final volume \ V2 = V1 + \frac{20}{100} \times V1 = 960 cc.

Understanding how gases expand with temperature changes is crucial in various fields, from weather prediction to engineering applications.

Temperature Conversion

Temperature conversion is essential for correctly applying gas laws, particularly in switching between Celsius and Kelvin. While Celsius is commonly used in daily life, Kelvin is often used in scientific calculations because it starts at absolute zero, making it an absolute measure for temperature.
To convert Celsius to Kelvin, simply add 273.15 to the Celsius temperature:

• \(T \text{(K)} = T \text{(°C)} + 273.15\).

In our exercise, we converted the initial temperature from 22°C to Kelvin:

• \(T1 = 22^{\text{°C}} = 22 + 273 = 295 \text{K}\).

After finding the final temperature in Kelvin, we converted it back to Celsius:

• \(T2 = 354 \text{K} - 273 = 81^{\text{°C}}\).

Understanding and accurately converting temperatures is key to solving problems involving gas laws efficiently.