Pearson Edexcel Physics

Unit 5: Thermodynamics, Radiation, Oscillations and Cosmology

5.3 Thermodynamics

Pearson Edexcel Physics Unit 5: Thermodynamics, Radiation, Oscillations and Cosmology 5.3 Thermodynamics Candidates will be assessed on their ability to::
125.	Be able to use the equations ΔE = mcΔθ and ΔE = LΔm
126.	CORE PRACTICAL 12: Calibrate a thermistor in a potential divider circuit as a thermostat
127.	CORE PRACTICAL 13: Determine the specific latent heat of a phase change
128.	Understand the concept of internal energy as the random distribution of potential and kinetic energy amongst molecules
129.	Understand the concept of absolute zero and how the average kinetic energy of molecules is related to the absolute temperature
130.	Be able to use the equation pV = NkT for an ideal gas
131.	CORE PRACTICAL 14: Investigate the relationship between pressure and volume of a gas at fixed temperature
132.	Be able to derive and use the equation [math]\frac{1}{2} m \langle c^2 \rangle = \frac{1}{2} k T[/math]

125) Be able to use the equations ΔE = mcΔθ and ΔE = LΔm
⇒ Specific Heat Capacity:
Transferring the same amount of heat energy to two different objects will increase their internal energy by the same amount.
However, this will not necessarily cause the same rise in temperature in both. How transferred heat energy affects the temperature of an object depends on three things:
– The amount of heat energy transferred
– The mass of the object
– The specific heat capacity of the material from which the object is made.
How much the temperature rises depends on the material, and is given by a property known as its specific heat capacity, c.
Different materials, and different phases of the same substance, have different specific heat capacities because their molecular structures are different.
This means that their molecules will be affected to different degrees by additional heat energy.
For a certain amount of energy, ΔE, transferred to a material, the change in temperature,Δθ, is related to the mass of material, m, and the specific heat capacity, c, by the expression:
[math]∆E = mc∆θ[/math]
With each quantity measured in SI units, this means that the specific heat capacity has units of [math]Jkg^{-1} K^{-1}[/math].
The change in temperature, Δθ, is the same whether measured in degrees celsius or kelvin, as the intervals are the same on both scales.
⇒ Specific latent heat:
To melt a solid that is already at the temperature at which it melts requires an input of additional energy.
This energy input does not raise the temperature, but is only used in breaking the bonds between molecules to change the state of the material.
The amount of additional energy needed to melt it is determined by two things:
– The mass of the solid substance
– The specific latent heat of fusion of the substance.
How much energy is needed to alter the molecular configuration depends on the material, and is given by a property known as its specific latent heat, L.
Different materials, and different phase changes of the same substance, have different specific latent heat values because their molecular structures are different.
This means that their molecules will be affected differently by additional heat energy.
For a certain mass of material, m, the amount of additional energy, [math]ΔE[/math], needed to make the phase change is related to the specific latent heat, L, by the expression:
[math]∆E = L∆m[/math]
With each quantity measured in SI units, this means that the specific latent heat has units of [math]Jkg^{-1}[/math]. The change in temperature is zero for any phase change, so the specific heat capacity need not be considered if the temperature is constant and only the state of matter changes.
⇒ Example:
How long will a 2 kW kettle take to raise the temperature of 800 grams of water from 20°C to 100 °C? The specific heat capacity of water is 4200[math]Jkg^{-1} K^{-1}[/math]
[math]\begin{gather}
\Delta \theta = 100 – 20 = 80^\circ\text{C} = 80\,\text{K} \\
\Delta E = mc\Delta \theta \\
\Delta E = (0.8)(4200)(80) \\
\Delta E = 268800\,\text{J} \\
t = \frac{E}{P} \\
t = \frac{268800}{2000} \\
t = 134.4\,\text{s}
\end{gather}[/math]
Therefore, the kettle takes 2 minutes and 14 seconds to heat this water to boiling temperature. We assume that no energy is wasted in heating the surroundings.
⇒ Example:
How long will a 150 W freezer take to freeze 650 grams of water which is already at 0°C? The specific latent heat of fusion of water is 334 000[math]Jkg^{-1}[/math]
[math]\begin{gather}
\Delta E = L \Delta m \\
\Delta E = (334000)(0.65) \\
\Delta E = 217100\,\text{J} \\
t = \frac{217100}{150} \\
t \approx 1447\,\text{s}
\end{gather}[/math]
Therefore, the freezer takes a little over 24 minutes to freeze this water. We assume that no energy is gained from the surroundings.
126) CORE PRACTICAL 12: Calibrate a thermistor in a potential divider circuit as a thermostat
Equipment:
– Thermistor
– Ohmmeter
– Kettle
– Alcohol thermometer
– Variable resistor
– Ice Beaker
Method:
– Install the ohmmeter across the thermistor and connect the resistor and thermistor in series to create a circuit.
– Carefully immerse the thermistor in the beaker filled with boiling water.
– Using a thermometer, note the starting temperature and the ohmmeter’s associated resistance.
– Add little amounts of ice gradually, mix, and then note the resistance and temperature changes.
– Continue until the water is below room temperature and all of the ice has been utilized.
Figure 1 Investigate how the motor (C) speed adjusts as the thermistor (A) temperature is altered. This circuit could be used to control a cooling fan.
Calculations:
– Create a calibration curve by plotting a resistance vs temperature graph.
– Once the resistance at the required switch-on temperature has been determined using the curve, use:
[math]V_{\text{out}} = V_{\text{in}} \left( \frac{R_1}{R_1 + R_2} \right)[/math]
– Construct a potential divider circuit, with the required second resistance to produce the desired [math]V_{out}[/math]
Safety Precautions:
– Take care when pouring boiling water.
– Don’t touch the beaker when the water temperature is high.
– Keep electrical connections away from the water, and clean up any spillages immediately.
127) CORE PRACTICAL 13: Determine the specific latent heat of a phase change
Investigating specific latent heat of fusion:
It is quite straightforward to measure the specific latent heat of fusion of water.
You can make a close measurement if you can measure the heat energy put into a known mass of ice, and if you set up the experiment well enough that almost all of the heat goes towards melting the ice.
A control funnel of crushed ice can also be used in which the heater is off, so we can see how much ice would have melted without the extra heating.
The difference in mass of water collected between the heated funnel and the control gives us the mass to use in the calculation.
Figure 2 Investigating the specific latent heat of fusion of ice.
Safety Precautions:
The heaters and metal blocks being tested will get hot enough to burn skin.
The immersion heaters must never have their tops under water as steam may be produced inside the heater which makes it explode.
128) Understand the concept of internal energy as the random distribution of potential and kinetic energy amongst molecules
129) Understand the concept of absolute zero and how the average kinetic energy of molecules is related to the absolute temperature
⇒ Internal Energy:
The average kinetic energy of the particles in a material gives the material its temperature.
However, in addition to this kinetic energy, each molecule will have some potential energy which is as a result of its position within the structure of the material, or in relation to other molecules in the substance.
This potential energy is due to the bonds between the molecules. If we sum the kinetic and potential energies of all the molecules within a given mass of a substance, we have measured its internal energy.
It is important to note that the molecules do not all have the same amount of kinetic and potential energies. This internal energy is randomly distributed across all the molecules, according to the Maxwell-Boltzmann distribution.
⇒ The Maxwell-Boltzmann Distribution:
If we identify the individual velocity of each molecule in a particular sample, the values will range from a few moving very slowly to a few moving very fast, with the majority moving at close to the average speed.
As they all have the same mass, their differences in kinetic energies are directly dependent on the speeds.
If we plot the kinetic energy against the number of molecules that have that energy, we obtain a curved graph called the Maxwell-Boltzmann distribution,
The characteristic shape of the graph in fig A shows that:
– there are no molecules with zero energy
– only a few molecules have high energy
– there is no maximum value for the energy a molecule can have.
A Maxwell-Boltzmann distribution graph is for one specific temperature. As the temperature changes, so the graph changes.
The peak on the graph moves towards higher energies (and therefore higher speeds) as the temperature increases.
Figure 3 The Maxwell-Boltzmann distribution of the speeds of a mixture of particles at one particular temperature.
⇒ Root-Mean Square Calculations:
There are two different ways of determining the average speeds of molecules in a material which are of interest to physicists.
The peak of the graph represents the kinetic energy value with the greatest number of molecules that have that energy.
Therefore, the speed corresponding to this kinetic energy is the most probable speed, co, if we were to select a molecule at random.
The second and more useful average is the root-mean-square speed, which has the symbol [math]\sqrt{\langle c^2 \rangle}[/math].
This is the speed associated with the average kinetic energy,[math]\frac{1}{2} m \langle c^2 \rangle[/math], where c is the speed of the particle, and m is its mass.
The root-mean-square speed, often abbreviated to rms speed, is found by squaring the individual speeds of a set of molecules, finding the mean of the squares, and then taking the square root of that mean.
⇒ Ideal Gases:
The three gas laws above were worked out from graphs of experimental results which showed distinct, straight best-fit lines, allowing scientists to claim them as empirical laws (laws worked out from experimental data).
However, if very accurate experiments are undertaken with a variety of different gases. we find that the laws are not perfectly accurate.
For example, Charles’s law suggests that if we reduce the temperature to zero kelvin, the gas has zero volume – it would disappear.
This does not happen, as the gas volume cannot reduce to less than the combined volume of its molecules.
The three gas laws are idealized, and would work perfectly if we could find a gas that did not suffer from the real-world difficulties that real gases have.
An ideal gas would have the following properties:
– The molecules have negligible size.
– The molecules are identical.
– All collisions are perfectly elastic and the time of a collision is significantly smaller than the time between collisions.
– The molecules exert no forces on each other, except during collisions.
– There are enough molecules so that statistics can be applied.
– The motion of the molecules is random. Assuming an ideal gas, we can combine the three gas laws to produce a single equation relating the pressure, volume, temperature and amount of a gas:
[math]pV = NkT[/math]
Where N is the number of molecules of the gas, and k is the Boltzmann constant. The temperature must be absolute temperature in kelvin.
This is known as the equation of state for an ideal gas, expressed in terms of the number of molecules present.
Sometimes, a more practically useful version can be used which refers to large quantities of gases, as we may find in a balloon, for example.
If we change from the Boltzmann constant, k, to using the gas constant for entire moles:
[math]pV = nRT[/math]
where n is the number of moles of the gas, and R is the universal gas constant, R = 8.31 [math]Jkg^{-1}mol^{-1}[/math]
⇒ The mole:
The mole (abbreviated to mol) is an SI unit used to measure the amount of a substance. One mole of a substance is defined to consist of Avogadro’s number of molecules of that substance.
[math]N_A[/math] is the symbol for Avogadro’s number [math]6.02 × 10^{23}[/math]
For one mole of an ideal gas:
[math]pV = RT[/math]
Comparing this with the equation of state:
[math]\begin{gather}
pV = N_A k T \\
N_A k = R \\
6.02 \times 10^{23} \times 1.38 \times 10^{-23} = 8.31
\end{gather}[/math]
130) CORE PRACTICAL 14: Investigate the relationship between pressure and volume of a gas at fixed temperature
⇒ Investigating Boyle’s Law:
Boyle’s law can be demonstrated using the apparatus shown in figure 4.
Measurements of the length of air which is trapped in the vertical glass column represent the volume of the gas, and the pressure is measured using the barometer.
A graph of pressure against 1/volume will give a straight best-fit line indicating that:
[math] p ∝ \frac{1}{V}[/math]
Figure 5 Graph of data from a Boyle’s law experiment. And Apparatus to demonstrate Boyle’s law.
Safety Precautions:
Wear eye protection and make sure all the connections are secure to avoid high pressure leaks. Reduce the pressure before disconnecting the pump.
131) Be able to derive and use the equation
[math]\frac{1}{2} m \langle c^2 \rangle = \frac{1}{2} k T[/math]
⇒ Average Molecular Kinetic Energy:
The ideal gas equation is:
[math]pV = NkT[/math]
and there’s an alternative equation from the kinetic theory of gases:
[math]pV = \frac{1}{3} N m \langle c^2 \rangle[/math]
Putting these together:
[math]\begin{gather}
\frac{1}{3} N m \langle c^2 \rangle = N k T \\
\frac{1}{3} m \langle c^2 \rangle = k T
\end{gather}[/math]
Multiply both sides by [math]\frac{3}{2}[/math]
[math]\begin{gather}
\frac{3}{2} \times \frac{1}{3} m \langle c^2 \rangle = \frac{3}{2} k T \\
\frac{1}{2} m \langle c^2 \rangle = \frac{3}{2} k T
\end{gather}[/math]
This derivation shows that it is a consistent consequence of the kinetic theory of gases.

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Pearson Edexcel Physics

Unit 5: Thermodynamics, Radiation, Oscillations and Cosmology

5.3 Thermodynamics

Pearson Edexcel Physics

Unit 5: Thermodynamics, Radiation, Oscillations and Cosmology

5.3 Thermodynamics

125) Be able to use the equations ΔE = mcΔθ and ΔE = LΔm

126) CORE PRACTICAL 12: Calibrate a thermistor in a potential divider circuit as a thermostat

127) CORE PRACTICAL 13: Determine the specific latent heat of a phase change

128) Understand the concept of internal energy as the random distribution of potential and kinetic energy amongst molecules

129) Understand the concept of absolute zero and how the average kinetic energy of molecules is related to the absolute temperature

130) CORE PRACTICAL 14: Investigate the relationship between pressure and volume of a gas at fixed temperature

131) Be able to derive and use the equation

[math]\frac{1}{2} m \langle c^2 \rangle = \frac{1}{2} k T[/math]

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