Question

Helium gas is in a cylinder that has rigid walls. If the pressure of the gas is 2.00 atm, then the root-mean-square speed of the helium atoms is \(\upsilon {_r}{_m}{_s}\) = 176 m/s. By how much (in atmospheres) must the pressure be increased to increase the \(\upsilon {_r}{_m}{_s}\) of the He atoms by 100 m/s? Ignore any change in the volume of the cylinder.

Short answer

The pressure must be increased by approximately 2.92 atm.

Step-by-step solution

Identify the given data

We are given the initial pressure of helium gas, \(P_1 = 2.00\) atm, and its initial root-mean-square speed, \(v_{rms1} = 176\) m/s. We need to find the increase in pressure such that \(v_{rms2} = v_{rms1} + 100 = 276\) m/s.

Understand the relationship between pressure and RMS speed

The root-mean-square speed \(v_{rms}\) of gas atoms is related to its temperature \(T\) by the equation \(v_{rms} = \sqrt{\frac{3kT}{m}}\), where \(k\) is the Boltzmann constant and \(m\) is the mass of the gas particles. However, considering the ideal gas law \(PV=nRT\), and since the volume \(V\) is constant in this problem, the pressure \(P\) is directly proportional to \(T\). Thus, \(v_{rms} \propto \sqrt{P}\).

Apply proportion relation

Using the relation \(v_{rms} \propto \sqrt{P}\), we can write: \(\frac{v_{rms2}}{v_{rms1}} = \sqrt{\frac{P_2}{P_1}}\). Substitute the known values: \(\frac{276}{176} = \sqrt{\frac{P_2}{2}}\).

Solve for \(P_2\)

Square both sides to remove the square root:\[\left(\frac{276}{176}\right)^2 = \frac{P_2}{2}\]Calculate \((\frac{276}{176})^2 = 2.46025\), thus:\[P_2 = 2 \times 2.46025 = 4.9205\] atm.

Determine the increase in pressure

The increase in pressure is given by \(\Delta P = P_2 - P_1\). Calculate:\[\Delta P = 4.9205 - 2.00 = 2.9205\] atm.

Key concepts

Ideal Gas Law

The ideal gas law is a fundamental principle used to understand the behavior of gases. It is expressed as \( PV = nRT \), where:

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

When dealing with problems involving gases, this equation helps relate these variables to each other.
In the exercise with helium gas, the volume \(V\) remains constant, simplifying the relationship between pressure and temperature.
This concept tells us that if the number of moles and the volume do not change, any increase in pressure is directly associated with an increase in temperature. This principle helps in calculating how the root-mean-square speed changes with pressure.

Pressure-Temperature Relationship

The connection between pressure and temperature in gases is crucial to understanding gas behavior. According to the ideal gas law, with constant volume and number of moles, the pressure of a gas is directly proportional to its temperature. If we double the temperature, the pressure also doubles, provided mass and volume remain unchanged.
In the context of the problem, increasing the root-mean-square speed of helium molecules requires raising the temperature, which in turn requires increasing the pressure due to their proportional relationship.
This understanding allows us to predict changes in gas behavior and calculate precise requirements, such as how much the pressure needs to be increased for a specific increase in the root-mean-square speed.

Helium Gas Properties

Helium is a noble gas with some unique properties that influence how it behaves as a gas. It is an inert, colorless, and odorless gas that is less dense than the air we breathe.
Here are some important properties of helium gas:

• Low atomic mass: Helium atoms have lower mass compared to many other gases, allowing them to move more quickly, as seen in their relatively high root-mean-square speeds.
• Non-reactive: Being a noble gas, helium does not easily form compounds with other elements, making it stable under various conditions.
• Low density and high internal energy: Its low density contributes to unique behavior in terms of buoyancy and thermal properties.

These attributes are critical when considering helium in applications where pressure, volume, and temperature are important, such as in this problem where the speed of its atoms relate directly to its pressure and temperature due to its distinct physical characteristics.