Question

Calculate the temperature at which \(\mathrm{rms}\) velocity of \(\mathrm{SO}_{2}\) molecules is the same as that of \(\mathrm{O}_{2}\) molecules at \(27^{\circ} \mathrm{C}\). Molecular weights of Oxygen and \(\mathrm{SO}_{2}\) are \(32 \mathrm{~g}\) and \(64 \mathrm{~g}\) respectively (A) \(327^{\circ} \mathrm{C}\) (B) \(327 \mathrm{~K}\) (C) \(127^{\circ} \mathrm{C}\) (D) \(227^{\circ} \mathrm{C}\)

Short answer

The temperature at which the rms velocity of SO2 molecules is the same as that of O2 molecules at 27°C is \(327^{\circ}\mathrm{C}\).

Step-by-step solution

Write down the expression for rms velocity for both gases

The expression for rms velocity (v_rms) is given by the formula: \(v_{rms} = \sqrt{\dfrac{3RT}{M}}\) Where: - R is the universal gas constant, - T is the temperature in Kelvin - M is the molar mass of the gas. For O2 molecules at 27°C or 300K (273 + 27), we can write the expression for v_rms as: \(v_{rms_{O2}} = \sqrt{\dfrac{3R \cdot 300}{32}}\) For SO2 molecules, we want to find the temperature where their rms velocity is equal to the rms velocity of O2 molecules. So we can write: \(v_{rms_{SO2}} = \sqrt{\dfrac{3R \cdot T_{SO2}}{64}}\) Now, we need to find the temperature \(T_{SO2}\) where the rms velocities for both gases are equal.

Set the rms velocities equal to each other and solve for the temperature

Since we want to find the temperature where the rms velocities for both gases are equal, we can set their expressions equal to each other: \(\sqrt{\dfrac{3R \cdot 300}{32}} = \sqrt{\dfrac{3R \cdot T_{SO2}}{64}}\) Now, we need to solve for \(T_{SO2}\) by squaring both sides and then isolating the variable on one side: \(\dfrac{3R \cdot 300}{32} = \dfrac{3R \cdot T_{SO2}}{64}\) Let's isolate \(T_{SO2}\) by multiplying both sides by 64, dividing by 3R, and simplifying: \(T_{SO2} = \dfrac{300 \cdot 64}{32} = 600\) So the temperature we found is in Kelvin. To convert it into Celsius, we need to subtract 273: \(T_{SO2(\mathrm{Celsius})} = 600 - 273 = 327^{\circ}\mathrm{C}\) The correct answer is option (A).

Key concepts

Molar Mass

Molar mass is a term used to describe the mass of one mole of a given substance. It is a critical component in chemical calculations and is usually expressed in grams per mole (g/mol). Each molecule of a compound is composed of different atoms, each contributing a specific mass. The molar mass is the sum of the masses of all atoms in a molecule. For example, for sulfur dioxide ( SO_2 ), the molar mass is 64 g/mol, since one mole of SO_2 contains one sulfur atom (32 g/mol) and two oxygen atoms (16 g/mol each).
Considering molar mass is essential in equations that involve gases, like calculating the root mean square ( RMS ) velocity, because it influences the speed at which gas molecules move. A lighter molecule typically moves faster than a heavier one at the same temperature.
For practical applications such as solving problems involving gas laws, knowing the molecular weight or molar mass allows you to accurately describe the behavior of gases under different conditions.

Universal Gas Constant

The universal gas constant, usually denoted as R , is an important value in chemistry and physics. It relates various physical properties of gases in calculations. The value of R is constant for all ideal gases, and it is crucial for numerous formulas, especially the ideal gas law. It is represented in units of energy per degree per mole, such as Joules per Kelvin per mole (J/K/mol).
In equations like the one used to determine the RMS velocity, R helps connect temperature and the molar mass to find how fast molecules move. Using the universal gas constant ensures the correct conversion of energy-related units into practical applications.
The presence of R in various equations makes it a universal tool for solving thermodynamic problems, supporting calculations for conditions of pressure, volume, and temperature, making it indispensable for understanding and predicting the behavior of gas molecules.

Temperature Conversion

Temperature conversion is a fundamental concept when working with scientific equations, especially those involving gases. In most scientific calculations, temperature needs to be in Kelvin because it is an absolute scale. The conversion from Celsius to Kelvin is straightforward: you simply add 273 to the Celsius temperature.
For instance, if you have a temperature of 27°C, converting it to Kelvin involves:

• Add 273 to the Celsius temperature: 27 + 273 = 300 K.

This conversion is essential since Kelvin provides a direct relationship to kinetic energy in a gas, thereby accurately reflecting changes involved in thermodynamic processes.
Understanding and applying temperature conversion correctly enables precise computations, influencing how formulas such as the ideal gas law are utilized and interpreted in practical scenarios.

Ideal Gas Law

The ideal gas law is a pivotal equation in chemistry and physics, described by the formula \( PV = nRT \), which connects pressure (P), volume (V), temperature (T), and moles of gas (n) through the universal gas constant (R). This equation provides relationships among various macroscopic properties of gases, hence explaining their behaviors under ideal conditions.
Real gases in practical situations might show deviations, but the ideal gas law offers a simplified model where gas particles are seen as infinitesimally small and non-interacting. This simplification simplifies a wide range of gas calculations.

• In our context, the law is vital in establishing the temperature parameter precisely in Kelvin for further computation steps.
• It also helps to understand how variations in one parameter affect others, maintaining a dynamic equilibrium between the gas properties.

Such understanding leverages how we approach problems involving gas velocities, pressures, and temperatures, allowing us to solve them efficiently.