Advertisements
Advertisements
Question
Derive Mayer’s relation.
Advertisements
Solution
- Consider one mole of an ideal gas that is enclosed in a cylinder by a light, frictionless airtight piston.
- Let P, V, and T be the pressure, volume, and temperature respectively of the gas.
- If the gas is heated so that its temperature rises by dT, but the volume remains constant, then the amount of heat supplied to the gas (dQ1) is used to increase the internal energy of the gas (dE). Since the volume of the gas is constant, no work is done in moving the piston.
∴ dQ1 = dE = CV dT ..............(1)
where CV is the molar specific heat of the gas at constant volume. - On the other hand, if the gas is heated to the same temperature, at constant pressure, the volume of the gas increases by an amount say dV. The amount of heat supplied to the gas is used to increase the internal energy of the gas as well as to move the piston backward to allow expansion of gas. The work is done to move the piston dW = PdV.
∴ dQ2 = dE + dW = Cp dT ..............(2)
Where CP is the molar specific heat of the gas at constant pressure. - From equations (1) and (2),
∴ Cp dT = CV dT + dW
∴ (Cp - Cv)dT = PdV ..............(3) - For one mole of gas,
PV = RT
∴ P dV = R dT, since pressure is constant.
Substituting equation (3), we get
(Cp - Cv) dT = R dT
∴ Cp - Cv = R
This is known as Mayer’s relation between CP and CV. - Also, CP = M0SP and CV = M0SV, where M0 is the molar mass of the gas and SP and SV are respective principal specific heats. Thus, M0SP - M0SV = R/J
Where J is the mechanical equivalent of heat.
SP - Sv = `"R"/("M"_0"J")`
RELATED QUESTIONS
Given below are observations on molar specific heats at room temperature of some common gases.
| Gas |
Molar specific heat (Cv) (cal mol–1 K–1) |
| Hydrogen | 4.87 |
| Nitrogen | 4.97 |
| Oxygen | 5.02 |
| Nitric oxide | 4.99 |
| Carbon monoxide | 5.01 |
| Chlorine | 6.17 |
The measured molar specific heats of these gases are markedly different from those for monatomic gases. Typically, molar specific heat of a monatomic gas is 2.92 cal/mol K. Explain this difference. What can you infer from the somewhat larger (than the rest) value for chlorine?
Define heat capacity and state its SI unit.
Name the S.I. unit of heat.
Find the time taken by a 500 W heater to raise the temperature of 50 kg of material of specific heat capacity 960 J kg-1K-1, from 18°C to 38° C. Assume that all the heat energy supplied by the heater is given to the material.
An electric heater of power 600 W raises the temperature of 4.0 kg of a liquid from 10.0℃ to 15.0℃ in100 s. Calculate:
- the heat capacity of 4.0 kg of liquid,
- the specific heat capacity of the liquid.
How does green house effect help in keeping the temperature of earth’s surface suitable for living of human beings?
Study the following procedure and answer the questions below:
1. Take 3 spheres of iron, copper and lead of equal mass.
2. Put all the 3 spheres in boiling water in a beaker for some time.
3. Take 3 spheres out of the water. Put them immediately on a thick slab of wax.
4. Note, the depth that each sphere goes into the wax.
i) Which property of substance can be studied with this procedure?
ii) Describe that property in minimum words.
iii) Explain the rule of heat exchange with this property.
(i) State whether the specific heat capacity of a substance remains the same when its state changes from solid to liquid.
(ii) Give one example to support your answer.
Define the term 'specific heat capacity' and state its unit.
(b) 2000 J of heat energy is required to raise the temperature of 4 kg of a
metal by 3°c. Which expression gives the specific heat capacity of the metal?
Define heat capacity.
State the condition for the flow of heat energy from one body to another.
Does the specific heat capacity of a substance depend upon its mass and rise in temperature only?
Specific heat capacity of substance A is 3.8 J g-1 K-1whereas the specific heat capacity of substance B is 0.4 J g-1 K-1. Which of the two is a good conductor of heat? How is one led to this conclusion?
A piece of ice is heated at a constant rate. The variation of temperature with heat input is shown in the graph below:
(i) What are represented by AB and CD?
(ii) What conclusion can you draw regarding the 110°c nature of ice from the above graph?
Calculate the amount of heat released when 5.0 g of water at 20°C is changed into ice at 0°C.
(Specific heat capacity of water = 4.2 J/g°C
Specific latent heat of fusion of ice = 336 J/g)
Write the name.
The amount of heat absorbed at constant temperature by unit mass of a liquid to convert into gaseous phase.
Explain how heat capacity of a solid can be determined by the method of mixture.
Derive Meyer’s relation for an ideal gas.
Which of the substances P, Q, or R has the lowest specific heat? The temperature v/s time graph is shown ______.
A monoatomic gas of pressure 'P' having volume 'V' expands isothermally to a volume '2V' and then adiabatically to a volume '16V'. The final pressure of the gas is ______.
(ratio of specific heats = `5/3`)
The molar specific heat of an ideal gas at constant pressure and constant volume is 'Cp' and 'Cv' respectively. If 'R' is the universal gas constant and the ratio 'Cp' to 'Cv' is 'γ' then CV = ______.
A metal ball of heat capacity 50J/°C loses 2000 J of heat. By how much will its temperature fall?
Which of the following substances (A, B and C) has the highest specific heat?
We would like to make a vessel whose volume does not change with temperature (take a hint from the problem above). We can use brass and iron `(β_(vbrass) = (6 xx 10^(–5))/K and β_(viron) = (3.55 xx 10^(–5))/K)` to create a volume of 100 cc. How do you think you can achieve this.
Find the odd one out:
Calculate the amount of heat energy required to raise the temperature of 200 g of copper from 20°C to 70°C. Specific heat capacity of copper = 390 J kg-1 K-1.
A 0.2 kg metal at 150°C is placed in a copper calorimeter (water equivalent 0.025 kg) with 150 cm³ water at 27°C. Final temperature is 40°C. Find the specific heat of the metal.
