ΔU is equal to
(a) isochoric work
(b) isobaric work
(c) adiabatic work
(d) isothermal work

Q: Why is $\Delta U = 0$ in an isothermal process?

For an ideal gas, the internal energy ($U$) depends solely on its absolute temperature ($T$). In an isothermal process, the temperature is kept constant ($\Delta T = 0$). Since the temperature doesn't change, the internal energy also doesn't change, making $\Delta U = 0$.

Q: What is the difference between heat ($q$) and internal energy ($U$)?

Internal energy ($U$) is the total energy stored within the system (kinetic and potential energy of molecules). Heat ($q$) is the energy in transit between the system and surroundings due to a temperature difference. You can think of internal energy as the bank balance and heat as a deposit or withdrawal.

Q: Why is work zero in an isochoric process?

Mechanical work done by or on a gas is given by $W = -p\Delta V$. In an isochoric process, the volume is constant, meaning the change in volume ($\Delta V$) is zero. Therefore, the work done is also zero.

Q: Can $\Delta U$ be negative?

Absolutely! $\Delta U$ is negative when the internal energy of the system decreases. This can happen if the system loses heat to the surroundings ($q < 0$) or if the system does work on the surroundings ($W < 0$, like in an adiabatic expansion).

Q: Is the First Law of Thermodynamics applicable to all processes?

Yes, the First Law is a fundamental law of nature. It is simply the law of conservation of energy applied to thermodynamic systems. It holds true for all processes, whether they are reversible, irreversible, isothermal, adiabatic, or anything else.

Question

$\Delta U$ is equal to

Options:

(A) isochoric work

(B) isobaric work

(C) adiabatic work

(D) isothermal work

Sigma Percentile · Accepted Answer

### The First Law of Thermodynamics: Unlocking the Secrets of Internal Energy

Imagine you have a sealed box containing a gas. This box is your thermodynamic system. The universe outside this box is the surroundings. The First Law of Thermodynamics is essentially the law of conservation of energy applied to this system. It tells us a beautiful and simple truth: energy cannot be created or destroyed, only transferred or transformed.

Mathematically, we express this as:

$$\Delta U = q + W$$

Here, **$\Delta U$** is the change in the internal energy of the system (the total kinetic and potential energy of all the molecules inside). **$q$** is the heat added to the system, and **$W$** is the work done *on* the system.

Let's explore how this master equation behaves under different constraints, which will lead us straight to the answer of our problem.

### The Isochoric Case: Locked Volume

What happens if we lock the walls of our box so they cannot move? This is an **isochoric process** (constant volume, $\Delta V = 0$).

Because the volume cannot change, the gas cannot expand to do work on the surroundings, nor can the surroundings compress the gas to do work on it. Therefore, the work done, $W$, is exactly zero.

Plugging this into our First Law equation:

$$\Delta U = q + 0 \implies \Delta U = q$$

In an isochoric process, any change in internal energy is purely due to heat exchange.

### The Isothermal Case: Locked Temperature

Now, imagine we place our box in a massive water bath that keeps its temperature perfectly constant. This is an **isothermal process** (constant temperature, $\Delta T = 0$).

For an ideal gas, the internal energy is a direct function of its temperature. If the temperature doesn't change, the internal energy cannot change. Therefore, $\Delta U = 0$.

Our First Law equation becomes:

$$0 = q + W \implies q = -W$$

This means any heat added to the system is entirely used by the system to do work on the surroundings.

### The Adiabatic Case: Perfect Insulation

Finally, let's wrap our box in a perfect thermal insulator, like thick styrofoam. No heat can enter or leave the system. This is an **adiabatic process** ($q = 0$).

Let's substitute this crucial condition back into the First Law equation:

$$\Delta U = 0 + W \implies \Delta U = W$$

This is a profound result! It tells us that in a perfectly insulated system, the only way to change the internal energy is by doing work on it (or letting it do work). If you compress the gas adiabatically, you do work on it ($W > 0$), and its internal energy increases (it gets hotter). If it expands adiabatically, it does work ($W < 0$), and its internal energy decreases (it gets colder).

### The Final Verdict

Returning to our original question: "$\Delta U$ is equal to...".

As we have just derived, $\Delta U$ is exactly equal to the work done, $W$, specifically under adiabatic conditions where $q = 0$. Therefore, $\Delta U$ is equal to **adiabatic work**.

This elegant deduction highlights the power of the First Law of Thermodynamics in understanding the physical world around us.

Animated Solution for Chemistry - Chemical Thermodynamics: $Δ U$ is equal to

Select Answer:

Visualized Solution

$First Law of Thermodynamics$

$Isochoric Process$

$Isothermal Process$

$Adiabatic Process$

$Conclusion$

The Sigma Insight: First Law of Thermodynamics

The First Law of Thermodynamics

The Isochoric Case

The Isothermal Case

The Adiabatic Case

The Final Verdict

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Animated Solution for Chemistry - Chemical Thermodynamics: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ΔU is equal to