Question

A gas has an initial volume of \(655 \mathrm{~mL}\) and an initial temperature of \(295 \mathrm{~K}\). What is its new temperature if volume is changed to \(577 \mathrm{~mL}\) ? Assume pressure and amount are held constant.

Short answer

The new temperature is approximately 260 K.

Step-by-step solution

Identify the Relationship

We are dealing with a gas and a change in its volume and temperature, while pressure remains constant. This indicates that we can use Charles's Law, which states that for a given mass of an ideal gas at constant pressure, the volume is directly proportional to its absolute temperature. Mathematically, this is expressed as \( V_1/T_1 = V_2/T_2 \).

Substitute Known Values

From the problem statement, we know \( V_1 = 655 \, \mathrm{mL} \), \( T_1 = 295 \, \mathrm{K} \), and \( V_2 = 577 \, \mathrm{mL} \). We need to find \( T_2 \). Substitute these values into the equation: \( \frac{655}{295} = \frac{577}{T_2} \).

Solve for Unknown Temperature

To solve for \( T_2 \), we first cross-multiply: \( 655 \times T_2 = 577 \times 295 \). Then, isolate \( T_2 \) by dividing both sides by 655: \( T_2 = \frac{577 \times 295}{655} \).

Calculate the New Temperature

Perform the multiplication and division to find \( T_2 \). First, multiply 577 by 295 to get 170215. Then, divide 170215 by 655, resulting in approximately \( 260 \mathrm{~K} \).

Key concepts

Ideal Gas

Charles's Law is an essential principle in understanding the behavior of gases, especially concerning ideal gases. An ideal gas is a theoretical concept where individual gas particles are assumed not to interact with one another aside from perfectly elastic collisions. This means that there are no attractions or repulsions between the particles, making calculations simpler and predictable.

In real-world applications, no gas fully behaves as an ideal gas due to interactions between molecules, but under certain conditions such as high temperatures and low pressures, many gases approximate ideal behavior closely. Hence, Charles's Law is often applied successfully in these scenarios.

• Ideal gas theory simplifies complex behaviors by ignoring intermolecular forces.
• Useful for modeling gas behaviors under various conditions.
• Allows for the study of relationships between pressure, volume, and temperature.

This framework is crucial for understanding the laws of gases, like Charles's Law, as it relies on these simplified conditions to explain gas behavior at a molecular level.

Direct Proportionality

In the world of gases, direct proportionality is a key concept, especially concerning Charles's Law. Direct proportionality between volume and temperature indicates that as one variable increases, the other increases by the same factor. In formulaic terms, it means the ratio of volume to temperature is constant when pressure is held constant.

This relationship is described by the equation: \[\frac{V_1}{T_1} = \frac{V_2}{T_2}\] Here, \(V_1\) and \(V_2\) are the initial and final volumes, while \(T_1\) and \(T_2\) are the initial and final temperatures respectively, all in Kelvin.

• Direct proportionality is reflected in the straight-line graph of volume vs. temperature.
• It shows unchanging conditions in terms of pressure and quantity of gas.
• The consistent ratio implies predictable changes in state variables.

This is a foundational element in understanding the predictable nature of gases under controlled conditions.

Volume-Temperature Relationship

The volume-temperature relationship outlined in Charles's Law reveals one of the fundamental aspects of gas behavior. When an ideal gas heats up at constant pressure, its volume expands. This is due to the increase in kinetic energy of the molecules, causing them to move faster and occupy more space.

In the classical Charles's Law experiment, if you double the temperature of a gas (in Kelvin), its volume theoretically doubles, provided the pressure and amount of gas remain unchanged. Conversely, cooling the gas would reduce its volume proportionally.

• This relationship is visually depicted in temperature versus volume graphs known as isobars.
• Understanding this relationship is essential for calculations involving thermal expansion of gases.
• It helps in predicting how a gas will behave when its temperature is altered.

For students, comprehending this concept is crucial for problem-solving in both theoretical and practical scenarios involving gases.