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# First Law of Thermodynamics

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The first law of thermodynamics is one of the three fundamental laws of thermodynamics. It is derived from the conservation of energy but is stated in a different and more useful manner for thermodynamics in order to include systems where the main method of energy transfer is heat transfer and work.

## The first law of thermodynamics: Equation

The first law of thermodynamics was derived in the 19th century by Rudolf Clausius and William Thomson. It states that the total change in the internal energy ΔU of a closed system is equal to the total heat transfer supplied into the system Q minus the total work done by the system W.

Figure 1. First law of thermodynamics. Source: Oğulcan Tezcan, StudySmarter.

### First law of thermodynamics: Internal energy (U)

The internal energy of a system on a smaller scale can be considered as the stored sum of the kinetic and potential energy of its atoms and molecules. However, it is more useful to define the internal energy on a larger scale, using the macroscopic quantities of a system, including pressure, temperature, and volume, to study the systems behaviour.

• Internal energy can be positive when heat is added to the system and/or work is done on the system.

• Internal energy can be negative when heat is removed from the system and/or work is done by the system on its surroundings.

### First law of thermodynamics: Heat (Q)

Heat (Q), measured in joules, is the energy that is transferred by molecular motion and collisions due to a temperature difference. When the system is taken as the reference, the heat that enters a system can be considered to be positive, while the heat exiting a system is negative.

### First law of thermodynamics: Work (W)

The work (W) of a system, measured in joules, is the energy that is transferred from one system to another or to its surroundings. This is a general form of mechanical work.

• When work is done by the system of reference, the work is defined as negative, as energy is lost from the system of reference and consumed by an external system or the surroundings.
• When work is done on the system of reference, the work is defined as positive, as energy is added to the system of reference and lost from an external system or the surroundings.

Some examples of positive and negative work depending on the chosen system of reference are given in the table below.

 Examples Work done on the system is positive Work done by the system is negative A steam engine produces work. The surroundings are the system of reference (energy is added to surroundings from the engine). The engine is the system of reference (energy is lost from the engine to the surroundings). Refrigerators consume work. Refrigerators are the system of reference (energy is added to refrigerators from the environment). The environment is the system of reference (energy is lost from the environment).

### First law of thermodynamics in differential form

The differential form of the first law of thermodynamics can be seen below. The differential form of the equation is used to describe in more detail the rate of change of heat and work and, as an extension, the rate of change of a systems internal energy.

$\partial U=\partial Q-\partial W$

In the case of the work done in a hydrostatic system, a system containing fluids, the differential equation can be simplified as shown below, where p is the pressure, and V is the volume of the system. Hence, the first law of thermodynamics can also be written as shown below when the volume of a fluid changes.

$\partial W=-p\partial V\partial U=\partial Q+p\partial V$

The negative sign indicates that the changes in volume are always opposite to the sign of the changes in work. For example, if work is positive, dV would be negative, and vice versa.

### First law of thermodynamics examples

The most common application of the first law of thermodynamics is the heat engine, which is used in trains, vehicles, etc.

Other applications include aeroplane engines, refrigerators systems, and heat pumps.

How much work is done by a gas that is compressed from 35L to 15L under a constant external pressure of 3 atm?

Solution:

$\partial W=-p\partial V=-p·\left({V}_{f}-V{}_{i}\right)$

As the gas is compressed, the work is positive, and dV is negative:

$\partial W=-3atm·\left(15L-35L\right)=60Latm$

As we need to convert this to Joules, we multiply by the gas constant J/mol K and divide by the gas constant 0.08206 L atm/mol K.

$\partial W=\frac{60atm·8.31447J/mol·K}{0.08206L·atm/mol·K}=6079J$

### First law of thermodynamics: Thermodynamic systems

There are three types of systems that can be observed in thermodynamics:

• Open systems, which exchange both energy and matter with their surroundings. For example, when boiling water in a pan, energy and matter are transferred from the pan to the surrounding atmosphere as steam.

• Closed systems, which exchange only energy with their surroundings. For example, a hot cup of coffee with a lid on transfers energy from the coffee to the surrounding atmosphere in the form of steam.

• Isolated systems, which are a special case of closed systems that transfer neither energy nor matter to other systems or their surroundings. For example, a perfectly insulated and closed nitrogen tank transfers neither energy nor matter to its surroundings.

## How does the first law of thermodynamics apply to gases?

Gases are sensitive to changes in macroscopic quantities such as volume, temperature, and density. For example, when the temperature increases, gases tend to expand due to the increased kinetic energy of the gas molecules. When the temperature decreases, the gases tend to compress. For constant pressure, the formula below can be used, where p is the pressure while Δv is the change in volume. The minus sign indicates that the work is done with respect to the system.

$W=-p·\Delta V$

From a thermodynamic perspective, the following apply:

• When a gas expands, the energy is transferred to the systems surroundings. Work is done by the gas on the surroundings. Here the work is negative (-W) with respect to the system (gas), as energy is released from the system.

• When a gas is compressed, energy is transferred from the surroundings to the gas. Work is done on the gas by the surroundings. Hence the work is positive (+ W) with respect to the system (gas).

• If the work done is being considered with respect to the surroundings, then the sign in the equation becomes positive. Work done becomes positive when the gas is expanded, while the work done is negative when the gas is compressed.

## First Law of Thermodynamics - Key takeaways

• Thermodynamics is the study of energy, heat, and temperature of matter.
• The first law of thermodynamics was derived from the conservation of energy theorem.
• The first law of thermodynamics states that the changes in internal energy are equal to the work done subtracted by heat addition.
• The most common application of the first law of thermodynamics is the heat engine.

The first law of thermodynamics states the relationship between the change in total internal energy of a system, the heat addition, and the work done. This can be mathematically expressed as ΔU = Q - W. Here, ΔU is the change in internal energy, Q is the heat added to the system, and W is work done by the system.

Heat (Q), work (W), and the change in total energy (ΔU).

The law of the conservation of energy.

Rudolf Clausius and William Thomson stated the first law of thermodynamics in 1865.

## Final First Law of Thermodynamics Quiz

Question

What is the study of thermodynamics?

It is a branch of physics that studies the behaviour of energy, heat, and the temperature of matter.

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Question

What is the equation of the first law of thermodynamics?

ΔU = ΔQ - ΔW

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Question

How many fundamental laws of thermodynamics are there?

3.

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Question

From which fundamental principle is the first law of thermodynamics derived?

The conservation of energy.

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Question

Who discovered the first law of thermodynamics?

William Rankin and Rudolf Clasius.

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Question

What is the internal energy in a system?

It is the sum of stored kinetic and potential energy of a system.

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What is heat (Q)?

It is the energy transferred by molecular motion and collisions due to a temperature difference.

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What is work (W)?

It is the energy transfer from one system to another.

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What is a closed system?

A closed system exchanges only energy but not matter.

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What is an open system?

A system that exchanges both energy and matter.

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Question

What is an isolated system?

A system that does not have any interactions with its surroundings.

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Question

What type of system is the following example: heating a saucepan on the stove with a lid on?

A closed system that exchanges energy, while the mass remains constant.

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Question

What type of system is the following example: heating a saucepan on the stove with no lid on?

An open system, as it exchanges both energy and mass.

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Question

What type of system is the following example: an ideal thermos containing a hot liquid?

An isolated system can be assumed to have constant energy and mass, and no external interactions.

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Question

Energy is added to a gas, causing it to expand at a pressure of 30 000 Pa from 0.3m3 to 0.6m3. How much work was done on the gas?

W=p DV= 30000(0.6-0.3)= 9000J.

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