Change in u Equation? Physics Formula Explained

This calculator works out the change in internal energy (ΔU) of a system using the first law of thermodynamics. You enter the heat added to the system (Q) and the work done by the system (W), and it returns the change in internal energy in joules. The formula is ΔU = Q − W, where a positive result means the system’s internal energy increased.

Formula

The first law of thermodynamics is a statement of energy conservation. In symbols:

ΔU = Q − W

• ΔU = change in internal energy (joules, J)
• Q = heat added to the system (joules, J)
• W = work done by the system (joules, J)

This sign convention is the most common in physics and engineering textbooks. Heat into the system is positive, work done by the system is positive. If work is done on the system (compression), W is negative, so the formula becomes ΔU = Q + |W|.

Worked Example

Using the default values: Q = 1500 J of heat added, W = 400 J of work done by the system.

1. Start with the formula: ΔU = Q − W
2. Substitute the numbers: ΔU = 1500 J − 400 J
3. Subtract: ΔU = 1100 J

The internal energy increases by 1100 J. The calculator shows 1100.00 J, matching the hand calculation.

When to Use This Equation

The first law applies to any thermodynamic process where a system exchanges energy with its surroundings. Common situations include:

• Heating a gas in a sealed container (constant volume)
• Compressing or expanding a gas in a piston (PV work)
• Chemical reactions in an open or closed vessel
• Phase changes like melting or boiling

It is used in physics, chemistry, and engineering to predict whether a system heats up, cools down, or does work. For example, if a gas expands rapidly and does work but receives no heat (adiabatic expansion), ΔU is negative, so the gas cools.

How to Read the Result

The sign of ΔU tells you the direction of energy change:

• Positive ΔU – internal energy increased; the system got hotter (if volume is constant) or stored more energy.
• Negative ΔU – internal energy decreased; the system cooled or released energy.
• Zero ΔU – energy in equals energy out; the process is cyclic or perfectly balanced.

Remember that ΔU depends only on the initial and final states, not the path taken. For an ideal gas, ΔU is proportional to the change in temperature: ΔU = n·Cv·ΔT, where n is moles, Cv is molar heat capacity at constant volume, and ΔT is temperature change in kelvin.

Common Mistakes and Unit Mix-Ups

The most frequent error is confusing the sign of work. If a problem says “work done on the system� (e.g., compressing a gas), you must use a negative W in the formula. Another mistake is mixing units: heat and work must both be in joules. If one is given in calories, convert first (1 cal = 4.184 J). Also, ensure temperature differences are in kelvin, not Celsius, when using ΔU = n·Cv·ΔT – the size of a kelvin is the same as a degree Celsius, so a difference of 10 °C equals a difference of 10 K.

Limitations of the Formula

The first law is always true for closed systems, but the simple ΔU = Q − W assumes:

• No mass crosses the boundary (closed system). For open systems, you need the steady-flow energy equation.
• Work is only due to volume change (PV work). Other forms like electrical or shaft work must be included separately.
• Kinetic and potential energy changes are negligible. If the system accelerates or lifts, those terms appear in the energy balance.

For real gases, internal energy also depends on pressure and volume, but the first law still holds – you just need more data to compute ΔU.

Frequently Asked Questions

What does ΔU stand for in physics?

ΔU is the change in internal energy of a system. It is the difference between the final internal energy and the initial internal energy, measured in joules. Internal energy includes the kinetic and potential energy of all the molecules in the system.

Why is the formula ΔU = Q − W and not Q + W?

It depends on the sign convention. The most common convention defines W as work done by the system. When a gas expands, it does positive work, losing energy, so you subtract it. If your textbook defines W as work done on the system, the formula becomes ΔU = Q + W. Always check the convention in your problem.

What if work is done on the system instead of by the system?

Then W is negative. For example, if 200 J of work is done on the gas (compression) and 100 J of heat is added, ΔU = 100 − (−200) = 300 J. The internal energy increases by 300 J.

Can ΔU be zero for a process?

Yes. In a cyclic process (the system returns to its initial state), ΔU = 0, so Q = W. Also, in an isothermal process for an ideal gas, ΔU = 0 because temperature is constant, so all heat added becomes work done.

How do I convert calories to joules for Q and W?

Multiply calories by 4.184 to get joules. For example, 500 cal = 500 × 4.184 = 2092 J. This conversion is exact because the calorie is defined in terms of the joule.

Does the equation work for liquids and solids?

Yes, the first law applies to any system. For solids and liquids, work is often negligible, so ΔU ≈ Q. That is why heating a metal block raises its internal energy almost entirely as heat.

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in q | Heat added (Q) | J | 1500 | 0
in w | Work done by system (W) | J | 400 | 0
out du | Change in internal energy (ΔU) | J | q - w | 2
out q_net | Net heat (Q) for ΔU=0 | J | w | 2