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# What is the mechanical equivalent of heat?

At the beginning of the 19th century, thermodynamics and mechanics were considered as two totally independent fields of science. Joule’s merit was to show that there is a connection between energy transfer through work and energy transfer through heat.

Joule also helped establish the Law of Conservation of Energy, which is the First Law of Thermodynamics. This law refers to the internal energy (U) of a system, where it indicates that its constancy can only be altered by the work and heat exerted by the system or on the system.

Historical aspects

### Water and heat

In the year 1792, Benjamin Thompson, Count Rumford, published in Philophical Transaction a set of experimental results that indicated a relationship between the friction experienced by water and the generation of heat. This statement produced a change in known ideas about heat.

### Mechanical work and heat

Later, the experiments of James Prescott Joule (1818-1889) on the equivalence of work and heat, contributed to the establishment of a kinetic theory that established a relationship between mechanical work and heat.

This contravened the caloric theory, which indicated that heat was a fluid that passed from one system to another, producing an increase in temperature.

In 1840, Joule established that the amount of heat produced in water by an electrical current was proportional to the electrical resistance and to the square of the electrical current (intensity).

Later, in 1842 von Mayer published the existence of a relationship between mechanical work and heat. However, this same relationship was published independently by Joule in 1843. That same year Jules published his value for the mechanical equivalent of heat. Meanwhile, Julius von Mayer did so in 1845, although it was pointed out that the experimental basis for his result was unconvincing.

In 1845, Joule published a work entitled “The Mechanical Equivalent of Heat,” a publication in which he stated a numerical value for the equivalent of 772.24 foot-force pounds (4.1550 joule · cal -1 ). These experiments showed a relationship between friction and generated heat.

In 1920 the value of the mechanical equivalent of heat was corrected to 4,186 J / g of water, this value being then defined as the amount of mechanical work necessary to vary the temperature of a gram of water from 14.5 ºC to 15.5 ºC.

In 1852, Joule and William Thompson discovered that when a gas expands its volume , without doing external work, its temperature drops. The so-called Joule-Thompson effect served as the basis for the establishment of a refrigeration industry in 19th-century England.

## Joule’s experiment

### Process

The experiment that allowed Joule to determine this equivalent consists of a copper container, which serves as a calorimeter, and in which a certain volume of water is placed.

The container has a lid that allows the insertion of a thermometer and a support for the paddles that are going to stir the water. The support consists of a crank and a spool of thread in which the threads that bind each of the two masses used in the experiment are incorporated.

Likewise, the part of the support that is immersed in the water is provided with paddles that serve to agitate it. Finally, the apparatus is provided with two rulers, one for each mass, with which the variation in their height is determined during the experiment.

As the masses fall, they rotate the support and the blades attached to it, producing an agitation of the water that translates into heat and an increase in temperature, as a consequence of the friction between the blades and the water.

By means of the crank, the masses are raised and the process is repeated several times, until there is an appreciable variation in temperature. The following video shows the operation of this experiment:

### Calculations

The mechanical work done when the two weights fall is the product of the loss of potential energy:

W = n · m · g · h (loss of potential energy when making masses)

Where n are the repetition of the fall of the masses, W the mechanical work to move the blades, m the masses of the blades, g the acceleration of gravity, and h the height traveled by the masses when falling.

The heat produced by the action of the paddles on the water, a consequence of the fall of the masses, is given by the expression:

Q = (M + W  ) (T 2 – T 1 )

Where Q is the heat produced, M the mass of the water, W ‘the water equivalent of the calorimeter, and T 2 – T 1 the temperature variation.

The mechanical equivalent of heat is then given by the relation:

J = W / Q

Which will be the same:

J = n · m · g · h / [(M + W ‘) · (T 2 – T 1 )]

= 4186 J / kcal

## Specific heat

### Thermal capacity of a substance

It is the amount of energy necessary to raise the temperature of a substance by 1 ºC:

C = Q / ΔT

Where C is the thermal capacity, Q the amount of heat absorbed, and ΔT the temperature variation.

### Specific heat of a substance

Specific heat is the heat capacity of a substance per unit mass:

Ce = Q / m Δt

Where Ce is the specific heat.

The specific heat of water (at 15ºC) is equal to 4.186 J / kg · ºC. Then, the value of the mechanical equivalent of heat corresponds to the value of the specific heat of water.

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