Internal energy is the sum of the randomly distributed kinetic energy and the potential energy of all particles in a body.

Kinetic energy can be:

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Translational

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Rotational

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Vibrational

The energy of a system can be increased by doing work on it or heating it.

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Doing work is the energy transfer due to a force

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Heating is a thermal energy transfer

A system can do work against an external force in some circumstances:

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\(CO_{2}\) will expand rapidly if released from a high pressure container

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Does work against the atmosphere, rapidly losing energy

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Cools enough to solidify to dry ice

The change in the internal energy of a system is equal to the sum of the energy transferred from or to the system by heating, and the energy transferred from or to the system as a result of work being done against or by an external force.

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Kinetic energy is directly related to temperature

Specific heat capacity, \(c\), is the energy required to raise the temperature of 1kg of a material by 1K without any change of state, measured in \(Jkg^{-1}K^{-1}\)

\begin{equation} Q=mc\Delta\theta\nonumber \\ \end{equation}\(c_{\text{water}}=4190Jkg^{-1}K^{-1}\); a fairly large value. Hence, a large body of water will take alot of energy to heat.

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Immersion heater in a block of a metal

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Time and temperature change recorded

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\(Q\) calculated as the energy transferred by the heater, using its current, voltage and power

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Water boiled to \(100^{\circ}C\) with object

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Object placed in cool water

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Temperature changes equated

If an object is dropped, and hits the ground without rebounding, all \(GPE\) will have been transferred to \(KE\), and then to internal energy. This leads on to:

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Lead shot in a tube

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Length of tube and temperature of lead recorded

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Tube repeatedly turned over, moving lead through length

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Total \(GPE\) changes calculated, and equated to new temperature of lead

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Gives a very rough estimate

Water is good for storing energy because of its high \(c\), so it is used as a heat transfer liquid. With flow calculations, it is often used to use:

\begin{gather*} \text{rate}=\frac{\Delta v}{\Delta t}=\frac{\mathrm{d}v}{\mathrm{d}t} \\ E=pt \\ \rho=\frac{m}{v} \\ \end{gather*}When a substance changes state, work has to be done to break the intermolecular bonds. Whilst this is happening, potential energy will increase whilst kinetic energy remaincs constant. Work often has to be done against the surroundings if an object is expanding.

The energy required to change 1kg of a liquid into 1kg of a gas with no change in temperature.

The energy required to change 1kg of a liquid into 1kg of a solid with no change in temperature. For a substance of mass \(m\), and specific latent heat of fusion \(l\), the energy transferred is given by

\begin{align} Q=ml,\therefore l=\frac{Q}{m}.\notag \\ \end{align}- •
Energy is supplied at a constant rate

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Temperature will rise till melting point

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Substance will melt at constant temperature

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Temperature of liquid will rise once all melted

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Liquid will boil at constant temperature

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Temperature of gas will rise once all melted

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Process can be recorded with a data logger

Cooling will not take place at a constant rate, as it is dependent on the temperature of the surroundings. Thermal energy will be dissipated from a substance that is cooling.

The volume of a fixed mass of gas at constant temperature is inversely proportional to its pressure.

\begin{gather*} pV=\text{constant} \\ p_{1}V_{1}=p_{2}V_{2} \\ \end{gather*}The volume of a fixed mass of gas at constant pressure is directly proportional to its kelvin temperature.

\begin{gather*} \frac{V}{T}=\text{constant} \\ \frac{V_{1}}{T_{1}}=\frac{V_{2}}{T_{2}} \\ \end{gather*}- •
Gives rise to isotherms on a pressure volume gas, each representing a different temperature

For a fixed mass of gas at constant volume, the pressure is directly proportional to the kelvin temperature.

\begin{gather*} \frac{p}{T}=\text{constant} \\ \frac{p_{1}}{T_{1}}=\frac{p_{2}}{T_{2}} \\ \end{gather*}Avogradro’s law states that in constant conditions, \(V\propto N\), where \(N\) is the number of molecules.

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All of above are empirical laws

Makes several assumptions:

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Gases have a large number of identical molecules

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Molecules themselves have a negligible volume

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Molecules possess no internal potential energy

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All molecules collide perfectly elastically

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Molecules move randomly, and in straight lines

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Molecules exert no forces on each other except in collisions

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Molecules spend significant more time travelling than colliding

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Newtons laws of motions can be applied to the molecules

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These are reasonable because:

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Brownian motion provides evidence for random motion

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Gases can be compressed, showing the molecules are far apart

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Gas molecules are not observed to lose temperature or slow, so collisions must be elastic

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Combining the empirical gas laws,

\begin{equation} pV=NkT\nonumber \\ \end{equation}where \(k=1.38\times 10^{-23}JK^{-1}\) and is the Boltzmann constant. Hence,

\begin{equation} pV=nRT\nonumber \\ \end{equation}where \(R=N_{A}k=8.31JK^{-1}mol^{-1}\), and is the molar gas constant. \(n\) is the number of moles, and

1 mole is defined as the number of atoms in 12 gramps of carbon-12, equal to \(6.02\times 10^{23}\)

Hence,

A gas will do work whilst expanding, unless in a vacuum.

\begin{equation} W=p\Delta V\nonumber \\ \end{equation}Brownian motion is the rapid, random movement of small particles.