Problem 80 - A Complete Resource Book in Chemistry for JEE Main

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) \(3.0 \mathrm{~m} / \mathrm{sec}\)

Step-by-Step Solution

Answer

The average velocity at \(927^{\circ} \mathrm{C}\) will be \(0.6\mathrm{~m/s}\).

Step 1: Understand the Temperature Change

Convert temperatures from Celsius to Kelvin. Since temperature in thermodynamics is calculated in Kelvin, we will convert both given temperatures. The formula for this is: \( T(K) = T(°C) + 273.15 \).For $$27^{\circ} \mathrm{C}$$: - \( T_1 = 27 + 273.15 = 300.15 \) K.For $$927^{\circ} \mathrm{C}$$: - \( T_2 = 927 + 273.15 = 1200.15 \) K.

Step 2: Use the Relationship Between Velocity and Temperature

The average velocity \( v \) of gas molecules is related to the temperature \( T \) by \[ v \propto \sqrt{T} \]This implies that the ratio of velocities is equal to the square root of the ratio of temperatures:\[ \frac{v_2}{v_1} = \sqrt{\frac{T_2}{T_1}} \]

Step 3: Calculate the Velocity Ratio

Plug in the values for the temperatures we found previously.\[ \frac{v_2}{v_1} = \sqrt{\frac{1200.15}{300.15}} \]Calculate this to find the ratio.

Step 4: Solve for the New Velocity

Now, calculate the value of \( v_2 \).\[ \frac{v_2}{0.3} = \sqrt{4} = 2 \]So, \( v_2 = 2 \times 0.3 \mathrm{~m/s} = 0.6 \mathrm{~m/s} \).

Key Concepts

Ideal Gas Behavior

In the realm of physics, particularly in the study of gases, the concept of an "ideal gas" is a simplified model that aids in understanding atmospheric behaviors. An ideal gas is composed of perfectly elastic collisions between pressure-less particles. These particles do not attract or repel each other, making calculations much easier as complicating inter-molecular forces are not present.
Real gases might somewhat differ from this model, but at normal atmospheric conditions—like room temperature and pressure—many real gases behave quite similarly to ideal gases.

• The behavior of ideal gases is modeled by the Ideal Gas Law: \[ PV = nRT \]where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(T\) is the temperature in Kelvin.
• Ideal gas behavior assumptions help in calculations involving gas velocities, such as those involving average molecular speed and rate of effusion.
• Key properties considered in ideal gas behavior include temperature, volume, pressure, and number of moles.

Temperature Conversion in Thermodynamics

In thermodynamics, temperature must often be converted into Kelvin from Celsius for calculations. Kelvin is the absolute temperature scale, used because it allows thermodynamic equations to work universally without giving negative temperatures.
To convert Celsius to Kelvin, simply add 273.15 to the Celsius temperature:

• For example, converting \( 27^{\circ}\)C: \( T(K) = 27 + 273.15 = 300.15 \) K.
• For \( 927^{\circ}\)C: \( T(K) = 927 + 273.15 = 1200.15 \) K.

Accurate temperature conversion is crucial for predicting how other properties—like pressure and volume—will change according to gas laws. Remember:- The Kelvin scale starts at absolute zero, which is theoretically the lowest possible temperature.- Unlike Celsius, Kelvin does not involve negative values, providing a more accurate and complete framework for scientific calculations.

Velocity-Temperature Relationship

Understanding the relationship between velocity and temperature helps in analyzing the dynamics of gas molecules. The kinetic theory of gases teaches us that the speed of gas molecules increases with temperature. This is because as the temperature rises, molecules gain kinetic energy, resulting in faster movement.

• The average velocity of gas molecules is proportional to the square root of the temperature, described by: \[ v \propto \sqrt{T} \]
• When comparing velocities at different temperatures, this relationship translates into: \[ \frac{v_2}{v_1} = \sqrt{\frac{T_2}{T_1}} \]
• This shows that even a small rise in temperature results in a noticeable increase in molecular velocity.

Thus, in practice, if a gas is initially at \(27^{\circ}\)C and its temperature is raised to \(927^{\circ}\)C, the molecular velocity will alter significantly, affecting everything from pressure in a closed container to the rate of diffusion through a membrane. Always bear in mind, increasing the temperature of a gas without changing the amount of gas can influence kinetic properties such as diffusion rates, pressure, and volume expansion.
By utilizing these relationships, we can solve for new conditions of gas behavior, predicting how molecules will react to temperature changes in the real world.