Question

Suppose ideal gas equation follows \(V P^{3}=\) constant, Initial temperature and volume of the gas are \(\mathrm{T}\) and \(\mathrm{V}\) respectively. If gas expand to \(27 \mathrm{~V}\), then temperature will become (A) \(9 \mathrm{~T}\) (B) \(27 \mathrm{~T}\) (C) (T/9) (D) \(\mathrm{T}\)

Short answer

(A) \(9 \mathrm{~T}\)

Step-by-step solution

Identify the initial conditions

The problem states that the initial temperature and volume of the gas are T and V, respectively.

Setup the alternative ideal gas equation

According to the problem, VP^3 = constant. Since we need to find the relation between the initial and final temperatures, we can express this equation as follows: \(V_1P_1^3 = V_2P_2^3\) Where \(V_1\) and \(P_1\) are the initial volume and pressure, and \(V_2\) and \(P_2\) are the final volume and pressure.

Use the constant proportionality to find the relation between initial and final temperatures

Since \(V_1P_1^3 = V_2P_2^3\), we know that the product of volume and the cube of pressure remains constant during the expansion. We can now express the final volume as 27V, where V is the initial volume: \((V_1P_1^3) = (27V)P_2^3\) We know that the initial volume is V, so we can divide both sides of the equation by V to find the ratio between the cube of the initial pressure and the cube of the final pressure: \(P_1^3 = 27P_2^3\) Now, from the ideal gas law, we know that \(PV = nRT\), where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature. So for our initial conditions, we can write: \(P_1V_1 = nRT_1\) and for our final conditions: \(P_2(27V_1) = nRT_2\)

Solve for the final temperature

Now, we can eliminate n and R from these equations, since they're constant for the gas throughout the process. So we can rewrite our equation like this: \(\frac{P_1V_1}{T_1} = \frac{P_2(27V_1)}{T_2}\) Now, we know that \(P_1^3 = 27P_2^3\), which means that \(P_2 = \sqrt[3]{\frac{P_1}{27}}\). Plugging this back into the equation above, we get: \(\frac{P_1V_1}{T_1} = \frac{\sqrt[3]{\frac{P_1}{27}}(27V_1)}{T_2}\) Now, we can cancel out \(P_1\) and \(V_1\) from both sides and we're left with: \(\frac{1}{T_1} = \frac{\sqrt[3]{\frac{1}{27}}}{T_2}\) Now, solving for \(T_2\), we get: \(T_2 = \frac{T_1}{\sqrt[3]{\frac{1}{27}}}\) This simplifies to: \(T_2 = 9T_1\) So, the final temperature will be 9 times the initial temperature, which corresponds to the option (A) \(9 \mathrm{~T}\).

Key concepts

Thermodynamics

Thermodynamics is a branch of physics that deals with the relationships between heat and other forms of energy. In essence, it involves understanding how different energy exchanges occur and how they affect temperature, volume, and pressure among other properties of a system.
Fundamentally, thermodynamics is concerned with how energy moves through a system:

• Energy conservation forms the backbone of the first law of thermodynamics, which states that energy cannot be created or destroyed, only transformed.
• The second law introduces the concept of entropy, a measure of disorder or randomness, highlighting how energy tends to spread out or disperse.

When applied to gases, thermodynamics describes how changes in temperature, pressure, and volume affect the energy state of the gas. Understanding this allows us to predict how a gas might behave under different conditions, essential for calculating changes such as those seen in gas expansion exercises.

Gas Expansion

Gas expansion occurs when a gas changes its volume due to temperature and pressure variations. This process can be described mathematically using equations derived from the ideal gas law. During expansion, the gas molecules move further apart, causing an increase in volume.
Key aspects of gas expansion include:

• Temperature increase: If a gas expands with energy input, its temperature may rise.
• Pressure changes: As volume increases, pressure tends to decrease if the temperature remains constant.

In our exercise, when the volume becomes 27 times the original, the temperature also increases significantly to maintain the equation's balance. This demonstrates how thermodynamic principles regulate such changes, ensuring the gas's energy state evolves consistently.

Pressure-Volume Relationship

The pressure-volume relationship, often described by Boyle's law and aspects of the ideal gas law, is a cornerstone concept in understanding gas behaviors. In an ideal gas scenario, pressure and volume are inversely related when temperature is held constant. This means:

• If volume increases, pressure decreases.
• Conversely, if volume decreases, pressure increases.

In the given problem, we recognize a modified form of the ideal gas relationship where the product of volume and the cube of pressure remains constant. This situation highlights the effect of extreme volume changes, such as the 27-fold increase, on pressure. Consequently, for the equation to stay balanced, the temperature must adjust, hence why the temperature ends up being 9 times its original value when the volume increases significantly.