Question

The average velocity of ideal gas molecules at \(27^{\circ} \mathrm{C}\) is \(0.3 \mathrm{~m} / \mathrm{sec}\). The average velocity at \(927^{\circ} \mathrm{C}\) will be (a) \(0.6 \mathrm{~m} / \mathrm{sec}\) (b) \(0.3 \mathrm{~m} / \mathrm{sec}\) (c) \(0.9 \mathrm{~m} / \mathrm{sec}\) (d) \(3.0 \mathrm{~m} / \mathrm{sec}\)

Short answer

The average velocity at \(927^{\circ} \mathrm{C}\) is \(0.6 \mathrm{~m} / \mathrm{sec}\).

Step-by-step solution

Understand the Relationship

The average velocity of gas molecules is directly proportional to the square root of the absolute temperature T. This is expressed by the formula: \( v \propto \sqrt{T} \).

Convert Temperatures to Kelvin

Convert the given temperatures from Celsius to Kelvin by adding 273 to each Celsius temperature. For \(27^{\circ}C\), the Kelvin equivalent is \(27 + 273 = 300K\). For \(927^{\circ}C\), it becomes \(927 + 273 = 1200K\).

Use Proportional Relationship

Using the relationship \( \frac{v_2}{v_1} = \sqrt{\frac{T_2}{T_1}} \), substitute the known values: \( v_1 = 0.3 \) m/s, \( T_1 = 300 \) K, and \( T_2 = 1200 \) K. This results in \( v_2 = v_1 \cdot \sqrt{\frac{T_2}{T_1}} \).

Calculate the New Velocity

Calculate \( v_2 \) using the equation from Step 3: \( v_2 = 0.3 \cdot \sqrt{\frac{1200}{300}} = 0.3 \cdot \sqrt{4} = 0.3 \cdot 2 = 0.6 \) m/s.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry that relates the pressure, volume, temperature, and number of moles of a gas. It is represented by the equation \( PV = nRT \). In this equation, \( P \) stands for pressure, \( V \) is volume, \( n \) is the amount of substance in moles, \( R \) is the ideal gas constant, and \( T \) is the absolute temperature measured in Kelvin. This law assumes that the gas being considered behaves ideally, which means the gas molecules occupy negligible space and have no interactions with each other. For an ideal gas, the collisions between molecules are perfectly elastic, and they move randomly.

• Pressure (\(P\)) is measured in atmospheres (atm) or pascals (Pa).
• Volume (\(V\)) is often measured in liters (L) or cubic meters (m³).
• The ideal gas constant (\(R\)) has a value of 8.314 J/(mol·K).

Understanding and utilizing the Ideal Gas Law allows us to predict how a gas will respond when conditions such as temperature and pressure change. It is fundamental in studies related to gases and their properties.

Temperature Conversion

To solve problems involving gas laws and kinetic theory, it's important to convert temperatures to the correct unit of measurement, especially Kelvin. Kelvin is the SI unit for temperature and is utilized in physics and chemistry because it starts at absolute zero, the theoretical lowest possible temperature. Adding 273 to a Celsius temperature converts it to Kelvin.Steps for Temperature Conversion:

• Add 273 to the Celsius temperature to obtain Kelvin.
• For instance, \( 27^{\circ}C \) becomes \( 300K \) and \( 927^{\circ}C \) becomes \( 1200K \).

Kelvin ensures that the temperature values are non-negative, which is essential when using formulas in gas laws, as negative temperatures can produce non-physical results in these equations.

Root Mean Square Velocity

Root Mean Square (RMS) velocity is a concept used in the kinetic theory of gases to describe the average speed of particles in a gas. It is derived from the square root of the average of the squares of the velocities of all the gas molecules. This measure considers all the molecular speeds in a sample of gas and calculates a type of average velocity.Key Points of RMS Velocity:

• It is represented by \( v_{rms} \) and calculated using the formula: \( v_{rms} = \sqrt{\frac{3RT}{M}} \), where \( R \) is the gas constant and \( M \) is the molar mass.
• RMS velocity increases with temperature since the kinetic energy of gas molecules is directly proportional to temperature.

Understanding RMS velocity allows us to predict how the velocity of gas molecules will change with environmental conditions, providing insights into diffusion rates and other dynamic processes of gases.

Proportional Relationships

Proportional relationships are crucial in understanding the behavior of gases. In the context of kinetic theory, certain properties of gases, such as molecular velocity, are directly proportional to the square root of the temperature, as well as inversely proportional to molar mass.Important Aspects of Proportional Relationships:

• Velocity \( v \) is proportional to \( \sqrt{T} \), which means \( v = k \cdot \sqrt{T} \) for some constant \( k \).
• If temperature increases, the velocity of gas molecules also increases, provided the mass remains unchanged.
• This can be applied to derive changes in velocity when temperature changes using formulas like \( \frac{v_2}{v_1} = \sqrt{\frac{T_2}{T_1}} \).

These relationships help predict outcomes in thermodynamic processes, illustrating the interconnectedness of temperature, velocity, and other gas properties.