At constant volume, 4 mol of an ideal gas when heated from 300 K to 500 K changes its internal energ

Q: Why do we use $C_V$ and not $C_P$ in this equation?

The problem explicitly states that the process occurs at 'constant volume'. By definition, the heat capacity measured under these conditions is $C_V$. Furthermore, for an ideal gas, the change in internal energy $\Delta U$ is always calculated using $C_V$, regardless of the process.

Q: Is there any work done by the gas during this heating process?

No! Work done by a gas is given by $W = P \Delta V$. Since the volume is constant, the change in volume $\Delta V$ is zero, which means absolutely no mechanical work is done.

Q: Does the internal energy of an ideal gas depend on its volume or pressure?

No, this is a fundamental property of ideal gases. The internal energy of an ideal gas depends *only* on its temperature. That's why the formula $\Delta U = n C_V \Delta T$ is universally applicable for ideal gases.

Q: Can I use temperatures in Celsius instead of Kelvin for this calculation?

Yes, but only because we are calculating a *difference* in temperature ($\Delta T$). A change of $1^\circ\text{C}$ is exactly equal to a change of $1\text{ K}$. However, it is always a safer habit to convert to Kelvin in thermodynamics to avoid errors in other formulas.

Q: What if the gas was not ideal? Would this formula still work?

If the gas were real (non-ideal), its internal energy would depend on both temperature and volume due to intermolecular forces. The simple relation $\Delta U = n C_V \Delta T$ would no longer be perfectly accurate, and we would need more complex equations of state.

Question

At constant volume, 4 mol of an ideal gas when heated from 300 K to 500 K changes its internal energy by 5000 J. The molar heat capacity at constant volume is ............

Sigma Percentile · Accepted Answer

### The Physical Setup

Imagine you are observing a rigid, closed container. Inside this container, we have trapped exactly **$4 	ext{ moles}$** of an ideal gas.

Because the container is rigid, its volume cannot change. This is what we call an **isochoric process**. When the volume is constant, the gas cannot expand or compress, which means it cannot do any mechanical work on its surroundings.

### Heating the Gas

Now, let's turn up the heat! As we supply thermal energy to the system, the temperature of the gas begins to rise. We observe the temperature increasing from an initial state of $300 	ext{ K}$ to a final state of $500 	ext{ K}$.

Because no work is being done, the First Law of Thermodynamics tells us that all the heat supplied goes directly into increasing the internal energy of the gas. The problem states that this change in internal energy, denoted as $\Delta U$, is exactly $5000 	ext{ J}$.

### The Master Equation

To find the molar heat capacity at constant volume, $C_V$, we need to connect it to the internal energy. The master equation that links these properties for an ideal gas is:

$$ \Delta U = n C_V \Delta T $$

Here, $n$ is the number of moles, and $\Delta T$ is the change in temperature. This equation is incredibly powerful because it holds true for an ideal gas regardless of the process, but it is directly derived from the definition of heat capacity at constant volume.

### Executing the Calculation

Let's carefully substitute our known values into the master equation. We plug in $5000$ for $\Delta U$, $4$ for $n$, and $(500 - 300)$ for $\Delta T$:

$$ 5000 = 4 	imes C_V 	imes (500 - 300) $$

First, we simplify the temperature difference inside the bracket. Subtracting $300$ from $500$ gives us $200 	ext{ K}$. Our equation now looks much cleaner:

$$ 5000 = 4 	imes C_V 	imes 200 $$

Next, we multiply the number of moles by the temperature change. Four times two hundred is $800$. Now, we isolate $C_V$ by dividing both sides by $800$:

$$ C_V = \frac{5000}{800} $$

### The Final Answer

Canceling out the zeros simplifies our fraction to $\frac{50}{8}$. Dividing $50$ by $8$ gives us our final result:

$$ C_V = 6.25 	ext{ J K}^{-1} 	ext{ mol}^{-1} $$

This value represents the amount of energy required to raise the temperature of one mole of this specific ideal gas by one Kelvin while keeping its volume constant.

Think about this: if the gas were allowed to expand at a constant pressure, we would use $C_P$ instead. Because the gas would do work while expanding, it would require *more* heat to achieve the same temperature rise, meaning $C_P$ is always greater than $C_V$!

Animated Solution for Chemistry - Chemical Thermodynamics: At constant volume, 4 mol of an ideal gas when heated from 300 K to 500 K changes its internal energy by 5000 J. The molar heat capacity at constant volume is ............

Enter Numerical Value:

Visualized Solution

$The Physical Setup$

$Heating the Gas$

$The Master Equation$

$Substituting Values$

$Simplifying the Bracket$

$Solving for C_{V}$

$Final Answer$

$The Way Forward$

The Sigma Insight: First Law of Thermodynamics

The Physical Setup

Heating the Gas

The Master Equation

Executing the Calculation

The Final Answer

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