Question

The temperature of an ideal gas is increased from \(27^{\circ} \mathrm{C}\) to \(127^{\circ} \mathrm{C}\), then percentage increase in \(\mathrm{v}_{\mathrm{rms}}\) is (A) \(33 \%\) (B) \(11 \%\) (C) \(15.5 \%\) (D) \(37 \%\)

Short answer

The percentage increase in the root mean square velocity (\(\mathrm{v}_{\mathrm{rms}}\)) is approximately 15.5%.

Step-by-step solution

Convert temperatures to Kelvin

Since the given temperatures are in Celsius, we need to convert them to Kelvin by adding 273.15 to each of them. For the initial temperature, \(T_1 = 27 ^{\circ} \mathrm{C}\) + 273.15 = 300.15 K For the final temperature, \(T_2 = 127 ^{\circ} \mathrm{C}\) + 273.15 = 400.15 K

Find the initial and final root mean square velocities

The formula for \(\mathrm{v}_{\mathrm{rms}}\) is \(\sqrt{\dfrac{3 RT}{M}}\). Since we are considering the same gas, we can focus on the temperature term. The initial root mean square velocity, \(\mathrm{v}_{\mathrm{rms1}}\) is proportional to \(\sqrt{T_1}\) The final root mean square velocity, \(\mathrm{v}_{\mathrm{rms2}}\) is proportional to \(\sqrt{T_2}\)

Compute the percentage increase in root mean square velocity

To calculate the percentage increase in \(\mathrm{v}_{\mathrm{rms}}\), we will use the following formula: Percentage increase = \(\dfrac{\mathrm{v}_{\mathrm{rms2}} - \mathrm{v}_{\mathrm{rms1}}}{\mathrm{v}_{\mathrm{rms1}}}\) × 100 % Since \(\mathrm{v}_{\mathrm{rms1}}\) and \(\mathrm{v}_{\mathrm{rms2}}\) are proportional to \(\sqrt{T_1}\) and \(\sqrt{T_2}\) respectively, we can write: Percentage increase = \(\dfrac{\sqrt{T_2} - \sqrt{T_1}}{\sqrt{T_1}}\) × 100 % Substituting the values of \(T_1\) and \(T_2\) obtained in Step 1: Percentage increase = \(\dfrac{\sqrt{400.15} - \sqrt{300.15}}{\sqrt{300.15}}\) × 100 %

Evaluate the percentage increase

Now, we can compute the percentage increase: Percentage increase ≈ \(\dfrac{20.004 - 17.324}{17.324}\) × 100 % Percentage increase ≈ 15.5 %

Choose the correct answer

We find that the percentage increase in \(\mathrm{v}_{\mathrm{rms}}\) is about 15.5 %. Thus, the correct answer is (C) 15.5 %.

Key concepts

Temperature Conversion

Converting temperature from Celsius to Kelvin is a simple, yet crucial step in physics problems. The Kelvin scale is pivotal for scientific calculations and is the standard unit of measurement for thermodynamic temperature. It begins at absolute zero, which is the point where all kinetic energy stops.
To convert Celsius to Kelvin, you simply add 273.15. This quick conversion is essential when dealing with the Ideal Gas Law and other thermodynamic equations. Here are a few points to remember:

• Add 273.15 to the Celsius temperature to get the Kelvin temperature.
• The Kelvin scale eliminates negative temperatures, which is beneficial in gas law calculations.
• Using Kelvin ensures consistency, as it is the SI unit for temperature.

In this exercise, converting 27°C and 127°C gives you 300.15 K and 400.15 K, respectively. These conversions allow further calculations involving gas laws to be accurate and effective.

Ideal Gas

The concept of an ideal gas is key when exploring thermodynamics. An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles.
The Ideal Gas Law demonstrates the relationship between pressure, volume, temperature, and the number of moles of an ideal gas. It is expressed as:\[ PV = nRT \]where:

• \(P\) is the pressure of the gas.
• \(V\) is the volume of the gas.
• \(n\) is the number of moles.
• \(R\) is the universal gas constant.
• \(T\) is the temperature in Kelvin.

In our exercise, the assumption that the gas is ideal means that its particles are not affected by intermolecular forces, allowing us to solely focus on temperature changes. When temperature increases, so does the kinetic energy of gas particles, impacting metrics such as the root mean square velocity.

Kinetic Theory of Gases

The kinetic theory of gases is a model that helps explain the behavior of gases, attributing their physical properties to the motion of individual molecules.
The theory states that the particles in an ideal gas move in a constant, random motion and that the effects of these collisions explain macroscopic properties like pressure and temperature. Root mean square velocity (\(v_{rms}\)) is a crucial concept here, indicating the speed of particles in a gas. It's calculated using the formula:\[ v_{rms} = \sqrt{\dfrac{3RT}{M}} \]where:

• \(R\) is the gas constant.
• \(T\) is the temperature in Kelvin.
• \(M\) is the molar mass of the gas.

In our exercise, with increasing temperature, \(v_{rms}\) rises due to intensified molecular activity, crucial for understanding changes in properties like the pressure of a gas.

Percentage Increase Calculation

Calculating percentage increases is a common mathematical technique used to quantify changes in a variable relative to its starting value. In this exercise, you determine the percentage increase in the root mean square velocity (\(v_{rms}\)) after a temperature change.
The formula for calculating percentage increase is:\[ \text{Percentage increase} = \dfrac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \times 100 \% \]In our context:

• The Old Value is the initial \(v_{rms}\), corresponding to the initial temperature \(\sqrt{T_1}\).
• The New Value is the final \(v_{rms}\), corresponding to the final temperature \(\sqrt{T_2}\).

By applying this formula with the temperatures converted into Kelvins, the exercise shows how \(v_{rms}\) increases by approximately 15.5% when the gas temperature rises from 27°C to 127°C. Calculating the percentage increase provides a clear measure of how much the velocity changes relative to its initial state.