Question

RMS velocity of a gas at STP cannot be calculated by the following expressions (1) \(\sqrt{\frac{3 P V}{M}}\) (2) \(\sqrt{\frac{3 R T}{M}}\) (3) \(\sqrt{\frac{3 P}{d}}\) (4) \(\sqrt{\frac{3 R T}{d}}\)

Short answer

1, 3, and 4

Step-by-step solution

Identify RMS velocity formula

The formula for the Root Mean Square (RMS) velocity of a gas is given by \( \text{RMS velocity} = \sqrt{\frac{3 R T}{M}} \), where \( R \) is the gas constant, \( T \) is the temperature in Kelvin, and \( M \) is the molar mass of the gas.

Check each given expression

We need to verify which expressions correctly calculate the RMS velocity. 1. \( \sqrt{\frac{3 P V}{M}} \) includes \( P \) (pressure) and \( V \) (volume), but the correct formula does not involve volume. This is not correct. 2. \( \sqrt{\frac{3 R T}{M}} \) matches our correct RMS velocity formula. This is correct. 3. \( \sqrt{\frac{3 P}{d}} \) includes \( P \) (pressure) and \( d \) (density), which are not in the correct formula. This is not correct. 4. \( \sqrt{\frac{3 R T}{d}} \) includes density \( d \), which is not in the correct formula. This is not correct.

Identify incorrect expressions

From the analysis, the incorrect expressions are \( \sqrt{\frac{3 P V}{M}} \), \( \sqrt{\frac{3 P}{d}} \), and \( \sqrt{\frac{3 R T}{d}} \).

Key concepts

Root Mean Square velocity

Root Mean Square (RMS) velocity is a measure of the average speed of gas molecules in a sample. It is important in understanding the behavior of gases. The correct formula for RMS velocity is given by: \[\text{RMS velocity} = \sqrt{\frac{3RT}{M}}\] where:

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

This formula allows us to calculate the average speed of gas molecules based on the temperature and molar mass.

STP (Standard Temperature and Pressure)

STP stands for Standard Temperature and Pressure, a reference point in scientific calculations to ensure consistency. At STP, the temperature is defined as 273.15 K (0°C) and the pressure is 1 atm (101.3 kPa). These conditions are used to simplify the behavior of gases and are crucial in calculations involving laws of gases. The RMS velocity at STP provides a standard way to compare the speeds of different gases under the same conditions.

ideal gas law

The ideal gas law is an equation of state that describes the behavior of an ideal gas. It combines several other gas laws and is given by: \[ PV = nRT \] where:

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

This law is essential for linking the macroscopic properties of gases such as pressure, volume, and temperature with the amount of gas in moles. Using this law, one can derive parameters like the RMS velocity.

gas constant

The gas constant, represented by the symbol \( R \), is a fundamental constant in many gas-related equations, including the RMS velocity and the ideal gas law. Its value is approximately 8.314 J/(mol·K). This constant helps in relating various properties of gases at the molecular level, making it possible to perform calculations involving energy, temperature, and volume. For example, in the formula for RMS velocity, the gas constant allows us to incorporate the temperature directly into the calculation.

molar mass

Molar mass is the mass of one mole of a substance, typically measured in grams per mole (g/mol). It is a key factor in many chemical and physical equations, especially those involving gases. In the context of RMS velocity, the molar mass of a gas determines how quickly its molecules move at a given temperature. Lighter molecules (lower molar mass) move faster, while heavier molecules (higher molar mass) move slower. Thus, knowing the molar mass is crucial for accurate calculation of the RMS velocity.