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# 10.4: Internal Energy

Learning Objectives

• Define and calculate changes in the internal energy of the system.

## Internal Energy

Internal Energy is defined as the sum of all the energy in a system. This energy can take the form or work or heat. We use the symbol ΔE to represent any changes in the internal energy of the system and account for these changes through work and heat

$\Delta E = w + q$

We know that if we have an exothermic reaction the system loses heat and the sign of q is negative. If we have an endothermic reaction heat is gained by the system and the sign of q is positive. Any work done by the system uses energy and the system loses energy, so the sign of w is negative. If work is done on the system the system gains energy and the sign of w is positive.

Energy Change Sign
q

Heat is gained by the system (endothermic) +q
Heat is lost by the system (exothermic) -q
w Work is done on the system (compression) +w
Work is done by the system (expansion) -w

By conservation of energy any change in the energy of the system must account for any energy the system gained and any energy the system lost.

Example $$\PageIndex{4}$$:

What is the change in internal energy for a gas if the gas gains 10 kJ of heat and expands, performing 5 kJ of work?

Solution

We can use

$\Delta E = q + w$

As the gas (our system) gained 10 kJ of heat, q is positive

$q = +10 \,kJ$

The system did work, using energy in the process, so the system lost energy and w is negative.

$w = -5\, kJ$

Putting these together

$\Delta E = q + w = 10 \,kJ + (-5 \,kJ) = 10\, kJ - 5 \,kJ = +5\, kJ$

So the system gained 5 kJ of energy overall.

Key Takeaways

• Any change in the internal energy of a system is the sum of the heat and the work.
• The overall change in the internal energy of a system will be the net total of any energy gained and any energy lost.

Exercises $$\PageIndex{1}$$

1. A gas is compressed, decreasing the volume, is the sign of the work positive or negative? Explain.
2. A gas expands, increasing the volume, is the sign of the work positive or negative? Explain.

3. A gas exchanges 22 kJ of energy with the surroundings in an exothermic reaction, what is the sign of q?

4. A gas exchanges 25 kJ of energy with the surrounding in an endothermic reaction, what is the sign of q?

5. A gas is heated with 36 J of energy and expands, doing 33 J of work, what is sign and value of ΔE?

6. A gas is cooled, losing 43 J of energy and is compressed using 22 J of work, what is the sign and value of ΔE?

7. A gas undergoes an endothermic process and expands, for this process |q| > |w|, what is the sign of ΔE?

8. A gas undergoes an exothermic process and is compressed, for this process |q| > |w|, what is the sign of ΔE?

9. For an endothermic process that does work, if |w| > |q| what is the sign of ΔE?

10. For an endothermic process that does work, if |q| > |w| what is the sign of ΔE?

11. For an exothermic process where work is done on the gas, if |w| > |q| what is the sign of ΔE?

12. For an exothermic process where work is done on the gas, if |q| > |w| what is the sign of ΔE?

Answers

1. As work was done on the gas the gas gained energy, so the sign of the work would be positive.
2.
3. As the reaction was exothermic heat left the system, the sign of q is therefore negative, q = -22 kJ

4.
5. As the gas was heated it gained energy, therefore q is positive and q = +36 J. The gas did work so it lost energy, therefore w is negative and w = -33 J. ΔE = q + w = +36 J + (-33 J) = +3 J.

6. ΔE = -65 J
7. As the process is endothermic q > 0, as the gas expanded it did work, so w < 0. As the magnitude of q (|q|) is greater than the magnitude of w (|w|) we have a positive ΔE, ΔE = +|q| + (-|w|) = q - w and |q| > |w|.

8. ΔE < 0
9. ΔE < 0

10.
11. ΔE > 0

12.
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