Question

The rms speed of gas molecules is given by (A) \(2.5 \sqrt{\left[M_{0} /(R T)\right]}\) (B) \(\left.2.5 \sqrt{[}(\mathrm{RT}) / \mathrm{M}_{0}\right]\) (C) \(\left.1.73 \sqrt{[}(\mathrm{RT}) / \mathrm{M}_{0}\right]\) (D) \(1.73 \sqrt{\left[M_{0} /(R T)\right]}\)

Short answer

The correct formula for the root mean square speed of gas molecules is approximately (C) \(1.73 \sqrt{\frac{RT}{M_0}}\). This is the closest approximation to the correct equation \(v_{rms} = \sqrt{\frac{3RT}{M_0}}\) among the options given.

Step-by-step solution

Identify the correct formula

We are given four options for the rms speed of gas molecules: (A) \(2.5 \sqrt{\frac{M_0}{RT}}\) (B) \(2.5 \sqrt{\frac{RT}{M_0}}\) (C) \(1.73 \sqrt{\frac{RT}{M_0}}\) (D) \(1.73 \sqrt{\frac{M_0}{RT}}\) Our task is to find out which of these options matches the correct equation: \(v_{rms} = \sqrt{\frac{3RT}{M_0}}\). Step 2 - Compare the options to the correct formula

Compare the options to the correct formula

Let's compare the given options with the correct rms speed equation: (A) \(2.5 \sqrt{\frac{M_0}{RT}}\) ≠ \(\sqrt{\frac{3RT}{M_0}}\) (B) \(2.5 \sqrt{\frac{RT}{M_0}}\) ≠ \(\sqrt{\frac{3RT}{M_0}}\) (C) \(1.73 \sqrt{\frac{RT}{M_0}}\) ≠ \(\sqrt{\frac{3RT}{M_0}}\) However, we can see that option (C) is just a numerical approximation of the correct equation: Option (C) can be written as follows: \(1.73 \sqrt{\frac{RT}{M_0}} = \sqrt{\frac{3.00 RT}{M_0}}\) which is approximately equal to the original equation given. (D) \(1.73 \sqrt{\frac{M_0}{RT}}\) ≠ \(\sqrt{\frac{3RT}{M_0}}\) Step 3 - Identify the correct option

Identify the correct option

Option (C) is the closest approximation to the correct equation, hence the correct formula for the root mean square speed of gas molecules is: \[ \boxed{\text{(C)}\ 1.73 \sqrt{\frac{RT}{M_0}}} \] Note that although this option is not an exact match, it is the closest approximation among the options given.

Key concepts

Root Mean Square Speed

The root mean square (rms) speed is a concept from kinetic theory that helps us understand the average velocity of gas molecules. It's derived from the idea that while individual molecules in a gas move with varying speeds, we can calculate a representative speed using statistical methods. The rms speed is significant because it accounts for all molecules and provides an average, weighted towards higher speeds. Mathematically, the rms speed of gas molecules is given by the equation:
\[v_{rms} = \sqrt{\frac{3RT}{M_0}}\]Here, \( R \) is the ideal gas constant, \( T \) is the absolute temperature of the gas, and \( M_0 \) is the molar mass of the gas molecules. Understanding this formula is crucial because it shows that rms speed depends on both the temperature and the molecular mass of the gas.

• Higher Temperature: Increases the rms speed, meaning molecules move faster.
• Higher Molecular Mass: Decreases the rms speed, meaning molecules move slower.

Ideal Gas Law

The ideal gas law is a fundamental principle used to describe how gases behave under different conditions. It relates four physical properties of a gas: pressure (\( P \)), volume (\( V \)), temperature (\( T \)), and the amount of gas in moles (\( n \)). This law is expressed as:
\[ PV = nRT \]Where \( R \) is the universal gas constant, used as a proportional factor to make the units consistent. This law is key when calculating rms speed, as it highlights the correlation between temperature and rms speed.Using the ideal gas law, you can:

• Predict how a gas will respond if any of its properties change.
• Relate the pressure and volume of a gas to its temperature and amount.
• Determine the influence of changing variables on the speed of gas molecules.

By understanding these relationships, you can apply them to find the average speed of molecules in a gas sample as dictated by kinetic theory.

Kinetic Theory of Gases

The kinetic theory of gases is an essential framework used to understand the behavior and properties of gases. This theory makes several assumptions about gases:

• Gas molecules are in constant random motion.
• Their collisions with each other and with the walls of the container are perfectly elastic, meaning there is no net loss of energy from these collisions.
• The molecules occupy negligible space compared to the volume of the container.
• There are no attractive or repulsive forces between the molecules.

These simplifications help derive important properties, such as pressure brought about by molecular collisions. In connection to rms speed, the kinetic theory sets the foundation for understanding how temperature influences molecular movement. According to the theory, temperature is directly related to the kinetic energy of gas molecules. Therefore, higher temperatures mean more kinetic energy and faster molecular speeds.

Molecular Mass

Molecular mass (often measured in units like grams per mole or kilograms per mole) significantly affects the rms speed of gas molecules. It represents the mass of a molecule and is a crucial factor in determining the dynamics of gas behavior as described in kinetic theory.
In the rms speed formula \( v_{rms} = \sqrt{\frac{3RT}{M_0}} \), the molecular mass is in the denominator under the square root. This means:

• Heavier Molecules: As the molecular mass increases, the denominator gets larger, reducing the rms speed. Thus, heavier gas molecules move slower compared to lighter ones.
• Lighter Molecules: Conversely, a lower molecular mass results in a higher rms speed, indicating that lighter molecules move faster.

Understanding molecular mass helps explain why gases like hydrogen, which have a low molecular mass, travel at higher speeds compared to heavier gases like sulfur hexafluoride. This understanding is vital in fields ranging from chemical engineering to meteorology.