Problem 12 - Algebra A Combined Function

Question

Charles's law states that if the pressure \(P\) stays the same, the volume \(V\) of a gas is directly proportional to its temperature \(T\). If a balloon is filled with 20 cubic meters of a gas at a temperature of \(300 \mathrm{~K},\) find the new volume if the temperature rises \(360 \mathrm{~K}\) while the pressure stays the same.

Step-by-Step Solution

Answer

The new volume is 24 cubic meters.

Step 1: Understand Charles's Law

Charles's Law can be represented by the equation \( V_1 / T_1 = V_2 / T_2 \), where \( V_1 \) and \( T_1 \) are the initial volume and temperature, and \( V_2 \) and \( T_2 \) are the final volume and temperature respectively.

Step 2: Identify Known Values

From the problem, we know the initial volume \( V_1 = 20 \) cubic meters, the initial temperature \( T_1 = 300 \; \mathrm{K} \), and the final temperature \( T_2 = 360 \; \mathrm{K} \). We need to find the final volume \( V_2 \).

Step 3: Apply Charles's Law Formula

Using the equation \( V_1 / T_1 = V_2 / T_2 \), substitute the known values: \( 20 / 300 = V_2 / 360 \).

Step 4: Solve for \( V_2 \)

Cross-multiply to solve for \( V_2 \): \( V_2 = (20 \times 360) / 300 \). Calculate \( V_2 \):\[ V_2 = \frac{7200}{300} = 24 \; \text{cubic meters} \]

Step 5: Conclusion

With the increase in temperature and constant pressure, the new volume of the gas in the balloon is 24 cubic meters.

Key Concepts

Direct Proportionality

In mathematics and science, direct proportionality is a foundational concept that explains how two variables relate through a constant ratio. When we say two variables are directly proportional, it means that as one variable increases, the other also increases at a constant rate. This indicates a linear relationship between the two.
Charles's Law is a prime example of direct proportionality. Here, the volume of a gas (V) is directly proportional to its temperature (T), provided the pressure is constant.

• If the temperature rises, the volume increases proportionally because their ratio remains constant.
• Conversely, if the temperature decreases, the volume similarly decreases.

The equation that describes this relationship is \( V \propto T \) or more specifically, \( V = kT \), where \( k \)is the constant of proportionality. This constant is determined by the initial conditions of the system such as the initial volume and temperature. By understanding direct proportionality, predicting changes in the physical world, like the behavior of gases, becomes much easier to grasp.

Gas Laws

Gas laws are a set of fundamental principles that describe how gases behave and interact under different conditions of pressure, temperature, and volume. These laws are crucial to understanding real-world applications in chemistry and physics.

The primary gas laws include:

• Boyle’s Law: Explores the inverse relationship between pressure and volume when temperature is constant. \( P_1V_1 = P_2V_2 \)
• Charles's Law: As already discussed, shows the direct proportionality between volume and temperature with constant pressure. It helps us predict how gases expand as temperatures rise.
• Avogadro's Law: Relates volume and the number of moles of gas, demonstrating that at constant temperature and pressure, volume and number of particles are directly proportional.

Each of these laws is a piece of the bigger puzzle that governs how gases operate in different environments and is essential for scientific endeavors that involve any gas-related process.

Volume and Temperature Relationship

The relationship between the volume of a gas and its temperature is a fascinating direct relationship as described by Charles's Law. This relationship is contingent on maintaining a constant pressure.

As the temperature of a gas increases, its particles move more rapidly and tend to spread out, causing the volume to increase. This is because the increased kinetic energy of gas particles at higher temperatures forces them to take up more space. Conversely, if the temperature decreases, the kinetic energy of the particles decreases, and thus, the gas occupies less space, resulting in a smaller volume.

• For prediction and calculation purposes, this relationship is expressed in the formula: \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \)
• This allows us to calculate unknowns, like predicting the new volume when temperature changes, as in the original exercise example.

Engaging with this relationship also helps us understand everyday phenomena — like why hot air balloons rise and why car tires might slightly deflate in cold weather.