Question

In another test, the valve of a 500-L cylinder full of the gas mixture at 2000 psi (gauge pressure) is opened wide so that the gas rushes out of the cylinder very rapidly. Why might some $N_{2}O$ condense during this process? (a) This is an isochoric process in which the pressure decreases, so the temperature also decreases. (b) Because of the rapid expansion, heat is removed from the system, so the internal energy and temperature of the gas decrease. (c) This is an isobaric process, so as the volume increases, the temperature decreases proportionally. (d) With the rapid expansion, the expanding gas does work with no heat input, so the internal energy and temperature of the gas decrease.

Short answer

Option (d) is correct: rapid expansion decreases temperature, causing _2O condensation.

Step-by-step solution

Understanding the Initial Condition

The valve of a 500-L cylinder is opened wide, allowing gas to escape rapidly. This process is taking place quickly, meaning that the expansion process is likely adiabatic (without heat exchange with the surroundings) due to the speed of the change.

Identify the Process Type

For gas expanding rapidly without exchanging heat (negligible time for heat exchange), it's often an adiabatic process rather than an isochoric, isobaric or another process. Options (a) and (c) mention isochoric and isobaric processes, which might not be accurate for rapid expansion.

Consider the Thermodynamics Involved

In an adiabatic process, the internal energy change is due to work done by the system. As gas expands, it does work on its surroundings, decreasing its internal energy. This would result in a decrease in temperature (as described in option (d)).

Analyzing Nitrous Oxide Condensation

When temperature decreases rapidly, a gas with a high boiling point like nitrous oxide ( _2O) might reach a temperature below its condensation point, leading it to change from a gaseous to a liquid state.

Conclusion from Options

Option (d) states that rapid expansion allows the gas to do work, decreasing internal energy and temperature without heat input, aligning with the adiabatic cooling description. This explains why some _2O could condense.

Key concepts

Adiabatic Process

In thermodynamics, an adiabatic process is one where no heat is transferred to or from the system. This can occur when the process happens so rapidly that there is no time for heat exchange. Adiabatic processes can occur in both compressions and expansions.

Imagine quickly opening the valve of a pressurized gas cylinder. The rapid expansion of the gas is likely to be adiabatic, as it happens fast enough that the system doesn't have time to exchange heat with its surroundings. Since no heat transfer occurs, changes in the internal energy of the gas are due solely to work done during expansion. This typically results in a drop in temperature as the gas uses its energy to expand.

Rapid Expansion

Rapid expansion is a swift change in the volume of gas, often resulting in significant thermodynamic effects. When gas expands rapidly, there is a decrease in pressure and volume. Such an expansion can cause the gas to do external work, which impacts its internal energy.

In rapid expansion, the gas molecules spread out quickly. This quick movement does not allow adequate time for heat to be added from the surroundings, essentially meaning that the expansion happens adiabatically. As a result, the internal energy of the gas decreases as its molecules perform work to spread the gas further. This phenomenon is key in understanding why certain gases may condense during such processes.

Nitrous Oxide Condensation

Nitrous oxide, or laughing gas, can condense under certain conditions of rapid temperature decrease. When a gas like nitrous oxide expands quickly and adiabatically, the temperature can drop to a point where condensation occurs, transitioning the gas into a liquid form.

Nitrous oxide has a relatively high boiling point compared to other gases. If the rapid expansion of the gas causes the temperature to fall below this boiling point, the gas can condense. This is particularly the case in scenarios where no heat enters the system to balance the loss of internal energy, leading to significant cooling. Understanding this concept is crucial, as it explains the behavior of gases under drastic thermodynamic changes.

Internal Energy Decrease

Internal energy is the total energy contained within a system, which includes both kinetic and potential energy at a molecular level. During a process like adiabatic expansion, the gas does work on its surroundings, reducing its internal energy.

Because no heat is added during an adiabatic process, as seen with the rapid expansion of gas, the system's internal energy decreases due to work done. This is reflected in a drop in temperature. The decrease in temperature can be so significant that it causes gases with certain characteristics, such as nitrous oxide, to condense. The change in internal energy is directly related to the work done, which highlights the interconnected nature of thermodynamic processes.