Question

The temperature at which the second virial coefficient of a real gas is zero is called: (a) critical temperature (b) eutactic point (c) boiling point (d) Boyle's temperature

Short answer

The correct answer is (d) Boyle's temperature.

Step-by-step solution

Understanding the terms

The second virial coefficient is a measure of the interaction between pairs of molecules in a gas sample. It's a part of the virial equation, which is used to account for intermolecular forces. The coefficient becomes zero when the repulsive and attractive forces between the molecules balance out.

Match the concepts

We need to match this concept with the terms provided. The critical temperature is the temperature above which a gas can't be liquefied, no matter the pressure applied. The eutactic point is a state at which all phases of a substance become identical with one another. The boiling point is the temperature at which a liquid changes state to become a gas. Boyle’s temperature is the temperature at which the second virial coefficient of a real gas is zero.

Choose the correct answer

As we have understood the meaning of each term, and now that we know that Boyle’s temperature is the point where the second virial coefficient is zero, we can confidently select (d) Boyle's temperature as the correct answer.

Key concepts

Second Virial Coefficient

Understanding the second virial coefficient is essential to grasping gas behavior beyond the ideal gas law. In simple terms, the second virial coefficient (\( B \) coefficient) quantifies the degree of interaction between pairs of molecules in a gas. As molecules come into proximity, they exert attractive and repulsive forces on each other.

When gases are studied under various conditions, their behavior often deviates from the predictions of the ideal gas law, especially under high pressure or at low temperatures. This is where the second virial coefficient comes in; it helps to correct the ideal gas law for intermolecular forces. The coefficient reflects the net effect of such forces. When the coefficient is positive, repulsive forces dominate, leading to a larger volume than anticipated. Conversely, a negative coefficient indicates that attractive forces are powerful, reducing the volume relative to an ideal gas.

Importantly, at Boyle's temperature, the second virial coefficient is zero; this means that the attractive and repulsive forces are in perfect balance, and the gas behaves most like an ideal gas at this point. Monitoring the changes in the second virial coefficient can provide valuable insights into molecular interactions within the gas.

Real Gas Behavior

The concept of real gas behavior extends from the idea that gases do not always follow the simple rules set out by the ideal gas law (\( PV = nRT \) where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is temperature). Real gases exhibit complex interactions definitively influenced by the conditions under which they are observed.

Intermolecular forces are key players in determining real gas behavior. These are the same forces that contribute to phenomena like condensation and boiling. For instance, molecules will follow the ideal gas law closely at high temperatures and low pressures, as their kinetic energies are sufficient to overcome intermolecular attractions. Under these conditions, the molecules are far apart and move too fast for the intermolecular forces to significantly affect their behavior.

However, when a gas is compressed (high pressure) or cooled (low temperature), these forces become significant. The gas may condense into a liquid or solid, or simply deviate from the ideal predictions. This is why the real behavior of gases is often modeled using corrections that account for volume excluded by molecule size and attractions between molecules.

Virial Equation

The virial equation provides a robust framework to understand the behavior of real gases. It's a series equation that corrects the ideal gas law by accounting for molecular interactions. The basic form of the virial equation is expressed as follows:

\[\begin{equation} P = \frac{nRT}{V - nb} - \frac{n^2a}{V^2} \end{equation}\]
where P represents pressure, V is volume, T is temperature, and n denotes the number of moles. The constants a and b are specific to each gas and are indicators of the gas's potential for intermolecular interactions.

This equation is significantly more accurate than the ideal gas law but can be complex to use due to the additional parameters. The virial equation can be expanded to include terms beyond the second virial coefficient (B), such as the third (C), fourth (D), and so on. Each additional coefficient accounts for increasingly complex molecular interactions that might occur in denser gases. For most practical purposes, the second virial coefficient is the most significant, but higher-order coefficients can become relevant under more extreme conditions.