Suppose an amount of heat ΔQ supplied to a substance changes its temperature from T to `T + ΔT`. We define heat capacity of a substance to be
`color{orange} {S = (Delta Q)/(Delta T)}` .................12.4
We expect ΔQ and, therefore, heat capacity S to be proportional to the mass of the substance. Further, it could also depend on the temperature, i.e., a different amount of heat may be needed for a unit rise in temperature at different temperatures.
To define a constant characteristic of the substance and independent of its amount, we divide S by the mass of the substance m in kg:
`color{orange} {s = s/m = (1/m) (Delta Q)/(Delta T)}` ......................12.5
s is known as the specific heat capacity of the substance. It depends on the nature of the substance and its temperature. The unit of specific heat capacity is `J kg^-1 K^-1`.
If the amount of substance is specified in terms of moles μ (instead of mass m in kg ), we can define heat capacity per mole of the substance by
`color{purple} {C = S/ mu = 1/ mu (DeltaQ)/(Delta T)}` .......................12.6
C is known as molar specific heat capacity of the substance. Like s, C is independent of the amount of substance. C depends on the nature of the substance, its temperature and the conditions under which heat is supplied.
The unit of C is `J mol^-1 K^-1`. As we shall see later (in connection with specific heat capacity of gases), additional conditions may be needed to define C or s. The idea in defining C is that simple predictions can be made in regard to molar specific heat capacities.
Table 12.1 lists measured specific and molar heat capacities of solids at atmospheric pressure and ordinary room temperature. We will see that predictions of specific heats of gases generally agree with experiment.
We can use the same law of equipartition of energy that we use there to predict molar specific heat capacities of solids. Consider a solid of N atoms, each vibrating about its mean position. An oscillator in one dimension has average energy of `2 × 1/2 k_BT = k_BT`. In three dimensions, the average energy is `3 k_BT`.
For a mole of a solid, the total energy is
`color{purple} {U = 3 k_BT × N_A = 3 RT}`
Now, at constant pressure, `color{green} {ΔQ = ΔU + P ΔV ≅ ΔU}`, since for a solid ΔV is negligible. Therefore,
`color{orange} { C = (DeltaQ)/(DeltaT) = (Delta U)/(Delta T) = 3 R}` ........................12.7
As Table 12.1 shows, the experimentally measured values which generally agrees with predicted value 3R at ordinary temperatures. (Carbon is an exception.) The agreement is known to break down at low temperatures.
`color{navy}bbul("Specific heat capacity of water")`
The old unit of heat was calorie. One calorie was earlier defined to be the amount of heat required to raise the temperature of 1g of water by `1°C`. With more precise measurements, it was found that the specific heat of water varies slightly with temperature. Figure 12.5 shows this variation in the temperature range 0 to `100 °C`.
For a precise definition of calorie, it was, therefore, necessary to specify the unit temperature interval. One calorie is defined to be the amount of heat required to raise the temperature of `1g` of water from 14.5 °C to 15.5 °C.
Since heat is just a form of energy, it is preferable to use the unit joule, J. In SI units, the specific heat capacity of water is `4186 J kg^1 K^1` i.e. `4.186 J g^-1 K^-1`. The so called mechanical equivalent of heat defined as the amount of work needed to produce `1 c a l` of heat is in fact just a conversion factor between two different units of energy : calorie to joule.
Since in SI units, we use the unit joule for heat, work or any other form of energy, the term mechanical equivalent is now superfluous and need not be used.
As already remarked, the specific heat capacity depends on the process or the conditions under which heat capacity transfer takes place. For gases, for example, we can define two specific heats : `color{lime} "specific heat capacity at constant volume"` and `color{lime} "specific heat capacity at constant pressure"`. For an ideal gas, we have a simple relation.
`color{orange} {C_p = C_v = R}` ...............12.8
where `C_p `and `C_v` are molar specific heat capacities of an ideal gas at constant pressure and volume respectively and R is the universalgas constant. To prove the relation, we begin with Eq. (12.3) for 1 mole of the gas :
`color{orange} {ΔQ = ΔU + P ΔV}`
If ΔQ is absorbed at constant volume, ΔV = 0
`color{purple} {C_v = ((DeltaQ )/(DeltaT) )_v = ((DeltaU)/(Delta T))_v = ((Delta U)/(Delta T ))}` ...................12.9
where the subscript v is dropped in the last step, since U of an ideal gas depends only on temperature. (The subscript denotes the quantity kept fixed.) If, on the other hand, ΔQ is absorbed at constant pressure,
`color{orange} {C_p = ((Delta Q )/(Delta T))_P = ((DeltaU)/(Delta T))_P + P ( (Delta V)/(Delta T))_P }` ..................12.10
The subscript p can be dropped from the first term since U of an ideal gas depends only on T. Now, for a mole of an ideal gas
`color{purple} {PV = RT}`
which gives
`color{navy} { P ((DeltaV)/(Delta T) )_P = R}` ..........12.11
Equations (12.9) to (12.11) give the desired relation, Eq. (12.8).
Suppose an amount of heat ΔQ supplied to a substance changes its temperature from T to `T + ΔT`. We define heat capacity of a substance to be
`color{orange} {S = (Delta Q)/(Delta T)}` .................12.4
We expect ΔQ and, therefore, heat capacity S to be proportional to the mass of the substance. Further, it could also depend on the temperature, i.e., a different amount of heat may be needed for a unit rise in temperature at different temperatures.
To define a constant characteristic of the substance and independent of its amount, we divide S by the mass of the substance m in kg:
`color{orange} {s = s/m = (1/m) (Delta Q)/(Delta T)}` ......................12.5
s is known as the specific heat capacity of the substance. It depends on the nature of the substance and its temperature. The unit of specific heat capacity is `J kg^-1 K^-1`.
If the amount of substance is specified in terms of moles μ (instead of mass m in kg ), we can define heat capacity per mole of the substance by
`color{purple} {C = S/ mu = 1/ mu (DeltaQ)/(Delta T)}` .......................12.6
C is known as molar specific heat capacity of the substance. Like s, C is independent of the amount of substance. C depends on the nature of the substance, its temperature and the conditions under which heat is supplied.
The unit of C is `J mol^-1 K^-1`. As we shall see later (in connection with specific heat capacity of gases), additional conditions may be needed to define C or s. The idea in defining C is that simple predictions can be made in regard to molar specific heat capacities.
Table 12.1 lists measured specific and molar heat capacities of solids at atmospheric pressure and ordinary room temperature. We will see that predictions of specific heats of gases generally agree with experiment.
We can use the same law of equipartition of energy that we use there to predict molar specific heat capacities of solids. Consider a solid of N atoms, each vibrating about its mean position. An oscillator in one dimension has average energy of `2 × 1/2 k_BT = k_BT`. In three dimensions, the average energy is `3 k_BT`.
For a mole of a solid, the total energy is
`color{purple} {U = 3 k_BT × N_A = 3 RT}`
Now, at constant pressure, `color{green} {ΔQ = ΔU + P ΔV ≅ ΔU}`, since for a solid ΔV is negligible. Therefore,
`color{orange} { C = (DeltaQ)/(DeltaT) = (Delta U)/(Delta T) = 3 R}` ........................12.7
As Table 12.1 shows, the experimentally measured values which generally agrees with predicted value 3R at ordinary temperatures. (Carbon is an exception.) The agreement is known to break down at low temperatures.
`color{navy}bbul("Specific heat capacity of water")`
The old unit of heat was calorie. One calorie was earlier defined to be the amount of heat required to raise the temperature of 1g of water by `1°C`. With more precise measurements, it was found that the specific heat of water varies slightly with temperature. Figure 12.5 shows this variation in the temperature range 0 to `100 °C`.
For a precise definition of calorie, it was, therefore, necessary to specify the unit temperature interval. One calorie is defined to be the amount of heat required to raise the temperature of `1g` of water from 14.5 °C to 15.5 °C.
Since heat is just a form of energy, it is preferable to use the unit joule, J. In SI units, the specific heat capacity of water is `4186 J kg^1 K^1` i.e. `4.186 J g^-1 K^-1`. The so called mechanical equivalent of heat defined as the amount of work needed to produce `1 c a l` of heat is in fact just a conversion factor between two different units of energy : calorie to joule.
Since in SI units, we use the unit joule for heat, work or any other form of energy, the term mechanical equivalent is now superfluous and need not be used.
As already remarked, the specific heat capacity depends on the process or the conditions under which heat capacity transfer takes place. For gases, for example, we can define two specific heats : `color{lime} "specific heat capacity at constant volume"` and `color{lime} "specific heat capacity at constant pressure"`. For an ideal gas, we have a simple relation.
`color{orange} {C_p = C_v = R}` ...............12.8
where `C_p `and `C_v` are molar specific heat capacities of an ideal gas at constant pressure and volume respectively and R is the universalgas constant. To prove the relation, we begin with Eq. (12.3) for 1 mole of the gas :
`color{orange} {ΔQ = ΔU + P ΔV}`
If ΔQ is absorbed at constant volume, ΔV = 0
`color{purple} {C_v = ((DeltaQ )/(DeltaT) )_v = ((DeltaU)/(Delta T))_v = ((Delta U)/(Delta T ))}` ...................12.9
where the subscript v is dropped in the last step, since U of an ideal gas depends only on temperature. (The subscript denotes the quantity kept fixed.) If, on the other hand, ΔQ is absorbed at constant pressure,
`color{orange} {C_p = ((Delta Q )/(Delta T))_P = ((DeltaU)/(Delta T))_P + P ( (Delta V)/(Delta T))_P }` ..................12.10
The subscript p can be dropped from the first term since U of an ideal gas depends only on T. Now, for a mole of an ideal gas
`color{purple} {PV = RT}`
which gives
`color{navy} { P ((DeltaV)/(Delta T) )_P = R}` ..........12.11
Equations (12.9) to (12.11) give the desired relation, Eq. (12.8).