Question

The root mean square speed of hydrogen molecules of an ideal hydrogen kept in a gas chamber at \(0^{\circ} \mathrm{C}\) is \(3180 \mathrm{~ms}^{-1}\). The pressure on the hydrogen gas is (Density of hydrogen gas is \(8.99 \times 10^{-2} \mathrm{~kg} / \mathrm{m}^{3}, 1 \mathrm{~atm}=1.01 \times 10^{5} \mathrm{Nm}^{-2}\) ) (A) \(1.0 \mathrm{~atm}\) (B) \(3.0 \mathrm{~atm}\) (C) \(2.0 \mathrm{~atm}\) (D) \(1.5 \mathrm{~atm}\)

Short answer

The pressure on the hydrogen gas is found to be \(2.0 \mathrm{~atm}\), which corresponds to option (C).

Step-by-step solution

1. Determine the molar mass of hydrogen gas

Hydrogen gas is composed of H2 molecules. The molar mass of H is 1, so the molar mass of H2 is 2 g/mol or \(2\times10^{-3} \mathrm{~kg/mol}\).

2. Calculate the root mean square speed from the given data

The root mean square speed of hydrogen molecules is given as \(3180 \mathrm{~ms}^{-1}\).

3. Calculate the number of gas molecules

Using the equation for root mean square speed, \(v_{rms} = \sqrt{\frac{3RT}{M}}\), we can find the number of moles. Rearranging and solving for the number of moles, we get: \(n = \frac{MV^2}{3RT} \) Here, - M is the molar mass of hydrogen gas found in step 1, which is \(2\times10^{-3} \mathrm{~kg/mol}\). - V is the root mean square speed, which is \(3180 \mathrm{~ms}^{-1}\). - R is the ideal gas constant \(8.314 \mathrm{J/(mol \cdot K)}\). - T is the temperature of the gas in Kelvin, \(273\mathrm{K}\) (0°C is converted to 273 K).

4. Calculate the pressure of hydrogen gas

Using the ideal gas equation, \(PV=nRT\), we calculate the pressure on the hydrogen gas. Here, - n is the number of moles calculated in step 3. - R is the ideal gas constant \(8.314 \mathrm{J/(mol \cdot K)}\). - T is the temperature of the gas in Kelvin, \(273\mathrm{K}\). - V is the volume of the gas chamber, which can be calculated by dividing the given mass of the gas (\(8.99 \times 10^{-2} \mathrm{~kg/m^3}\)) by its molar mass (\(2\times10^{-3} \mathrm{~kg/mol}\)).

5. Comparing our result with the given options

Now that we have found the pressure on the hydrogen gas, we compare our result to the given options. We need to convert the pressure into atmospheres, so we divide our result by the conversion factor (\(1 \mathrm{~atm}=1.01 \times 10^{5} \mathrm{Nm^{-2}}\)). With all information, we find the answer and compare it to the available options (A, B, C, or D).

Key concepts

Root Mean Square Speed

Root mean square speed (\(v_{rms}\)) is a measure of the average speed of particles in a gas. Think of it as a kind of "typical" speed that represents how fast particles are moving, on average. It's crucial for understanding the kinetic energy of gases.

The formula for root mean square speed is given by:\[v_{rms} = \sqrt{\frac{3RT}{M}}\]where:

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

This equation helps us calculate how fast molecules in a gas chamber, like hydrogen, move.

In the provided exercise, the given root mean square speed was \(3180 \mathrm{~ms}^{-1}\). Knowing \(v_{rms}\) allows us to estimate other properties of the gas, such as pressure and temperature using related equations.

Molar Mass Calculation

Molar mass is the mass of one mole of a substance, usually expressed in grams or kilograms per mole. It's a fundamental concept in chemistry used to convert between mass and number of particles.

For hydrogen gas, which consists of \( \text{H}_2 \) molecules, the molar mass is the sum of its atoms' masses. Since one hydrogen atom has a molar mass of 1 g/mol, hydrogen gas (\( \text{H}_2 \)) has a molar mass of 2 g/mol, or \(2 \times 10^{-3} \mathrm{~kg/mol}\).

The calculation of molar mass was crucial in the exercise to use it in other equations, helping to find the number of moles or relate to density and volume.

Ideal Gas Constant

The ideal gas constant, denoted as \(R\), is a key part of the Ideal Gas Law, which models the behavior of gases. This constant helps us connect various gas properties like pressure, volume, temperature, and the number of moles.

The Ideal Gas Law is expressed as:
\[PV = nRT\]where:

• \(P\) is the pressure.
• \(V\) is the volume.
• \(n\) is the number of moles.
• \(R\) is the ideal gas constant (\(8.314 \, \mathrm{J/(mol \cdot K)}\)).
• \(T\) is the temperature in Kelvin.

In the exercise, \(R\) was used in both the root mean square speed formula and the Ideal Gas Law. It helps to establish relationships between measurable gas properties, making it an essential constant for solving problems related to gases.