SPECIFIC HEAT

When heat energy flows into a substance, the temperature of the substance usually rises.

The heat required to raise the temperature of unit mass of a body through 1 ^oC or ( 1 ^oK ) is called specific heat capacity or simply specific heat of the material of the body. If $\Delta$Q heat changes the temperature of mass m by $\Delta$T.

$c=\frac{1}{m}\cdot \frac{\Delta Q}{\Delta T}$ … (1)

The SI unit of specific heat is J/kg K. Heat is so frequently measured in calories, therefore the practical unit cal/g C is also used quite often. The specific heat capacity of water is approximately 1 cal/g °C.

From Eq. (1), we can define the specific heat of a substance as "the amount of energy needed to raise the temperature of unit mass of that substance by 1°C (or 1 K)". A closely related quantity is the Molar heat capacity C. It is defined as,

$C=\frac{1}{n}\cdot \frac{dQ}{dT}$ … (2)

where n is the number of moles of the substance. If M is the molecular mass of the substance, then n = $\frac{m}{M}$ were m is the mass of the substance and,

C = $\frac{M}{m}\cdot \frac{dQ}{dT}$ … (3)

The SI units of C is J / mole K.

Key points:

(a) It depends on nature of material of body. Dulong and petit has found formula for elemental solids that (with few exceptions such as carbon)

Atomic weight x Specific heat = 6 cal / ^oC

So, heavier the element lesser will be the specific heat, i.e., C_Hg < C_Cu< C_Al

(b) Specific heat of a substance also depends on temperature (particularly at low temperatures) the variation of specific heat with temperature for wateris shown in Fig (A) for metals in Fig (B). This temperature dependence of specific heat is usually neglected.

(c) Specific heat also depends on the state of substance, i.e., solid, liquid or gas. e.g., specific heat of solid copper will be different from that of liquid copper. In case of water

(d) If a substance is undergoing change of state which takes place at constant temperature (called isothermal change) , specific heat

$c=\frac{1}{m}\cdot \frac{dQ}{dT}$ = $=\frac{1}{m}\cdot \times \frac{dQ}{0}$= $\infty$ [as $\Delta$T = 0]

i.e., specific heat of a substance at its melting pint or boiling point or isothermal change is infinite.

(e) Specific heat is found to be maximum for hydrogen (3.5 cal/gm °C) then for water (1 cal/gm °C = 4200 J/kg K). For all other substances specific heat is lesser than 1 cal/gm °C and is minimum for radon and actinium (= 0.22 cal/gam °C)

(f) If the temperature of a body changes without transfer of heat with the surroundings (adiabatic change) as in shaking a liquid or compressing a gas,

c = $\frac{Q}{m\Delta T}=\frac{0}{m\Delta T}$= 0 [as Q = 0]

i.e., specific heat of a substance, when it undergoes adiabatic change, is zero.

(g) Specific heat of a substance can also be negative. Negative specific heat means that in order to raise the temperature, a certain quantity of heat is to be withdrawn from the body. Specific heat of saturated water vapours is negative.

(h) When specific heats are measured, the values obtained are also found to depend on the conditions of the experiment. In general measurements made at constant pressure are different from those at constant volume. For solids and liquids this difference is very small and usually neglected. The specific heat of gases are quite different under constant pressure condition (c_p) to constant volume condition (c_v).

(i) As by definition c = (Q/m $\Delta$T), heat required to change the temperature of m gm of a substance through $\Delta$T:

Q = mc $\Delta$T

and as $\Delta$T = (Q/mc), greater the specific heat of a substance lesser will be the change in temperature for a given mass when same amount of heat is supplied. Now as specific heat of water is very large (1 cal/g °C), by absorbing or releasing large amount of heat its temperature changes by small amounts. This is why, it is used in hot water bottles or as coolant in radiators. This is also how the sea moderates the climate of nearby coastal land.

WATER EQUIVALENT

Water–equivalent of a body is the mass of water which when given same amount of heat as to the body, changes the temperature of water through same range as that of the body, i.e.,

W = (m x c) gm

The unit of water equivalent W is gm while its dimensions [M]. Units and dimensions of some physical–quantities used in heat are given below in a tabular form.

MEASUREMENT OF SPECIFIC HEAT CAPACITY & ERROR ANALYSIS

As shown in the figure Regnault's apparatus to determine the specific heat capacity of a solid heavier than water, and insoluble in it. A wooden partitions P separates a steam chamber O and A calorimeter C. The steam chamber O is a double walled cylindrical vessel. Steam can be passed in the space between the two walls through an inlet A and it can escape out through an outlet B. The upper part of the vessel is closed by a cork. The given solid may be suspended the vessel is closed by a cork. The given solid may be suspended in the vessel by a thread passing through the cork. A thermometer T₁ is also inserted into the vessel to record the temperature of the solid. The stem chamber is kept on a wooden platform with a removable wooden disc D closing the bottom hole of the chamber. To start with, the experimental solid (in the form of a ball or a block) is weighed and then suspended in the steam chamber. Steam is prepared by boiling water in a separate boiler and is passed through the steam chamber. A calorimeter with a stirrer is weighed and sufficient amount of water is kept in it so that the solid may be completely immersed in it. The calorimeter is again weighed with water to get the mass of the water. The initial temperature of the water is noted.

When the temperature of the solid becomes constant (say for 15 minutes), the partition P is removed. The calorimeter is taken below the steam chamber, the wooden disc D is removed and the thread is cut to drop the solid in the calorimeter. The calorimeter is taken to its original place and is stirred. The maximum temperature of the mixture is noted.

Calculation:

Let the mass of the solid =m₁

mass of the calorimeter and the stirrer =m₂

mass of the water =m₃

specific heat capacity of the solid =s₁

specific heat capacity of the material of the calorimeter(and stirrer) =s₂

specific heat capacity of water =s₃

initial temperature of the solid =$\theta$₁

initial temperature of the calorimeter, stirrer and water =$\theta$₂

final temperature of the mixture =$\theta$

We have

heat lost by the solid =m₁s₁($\theta$₁-$\theta$)

heat gained by the calorimeter (and the stirrer) =m₂s₂(-$\theta$₂)

Assuming no loss of heat to the surrounding the heat lost by the solid goes into the calorimeter stirrer and water. Thus

m₁s₁($\theta$₁ – $\theta$) = m₂s₂ ($\theta$–$\theta$₂)+ m₂s₃($\theta$ – $\theta$₂) ……(1)

or s₁ = $\frac{(m_2{s}_2+m_3{s}_3)(\theta -{\theta }_2)}{m_1({\theta }_1-\theta )}$

Knowing the specific heat capacity of water (s₃ = 4186 J/kg-K) and that of the material of the calorimeter and the stirrer (s₂=389 J/kg-K if the material be copper), one can calculate s₁.Specific heat capacity of a liquid can also be measured with the Regnault apparatus. Here a solid of known specific heat capacity s₁ is used and the experimental liquid is taken in the calorimeter in place of water. The solid should be denser than the liquid. Using the same procedure and with the same symbols we get an equation identical to equation (1) above, that is,

m₁s₁($\theta$₁– $\theta$) = m₂s₂ ($\theta$–$\theta$₂) + m₃s₃($\theta$ – $\theta$₂)

in which s₃ is the specific heat capacity of the liquid.

We get

S₃ = $\frac{m_1{s}_1+({\theta }_1-\theta )m_2{s}_2}{m_3(\theta -{\theta }_2)m_3}$

Error analysis

After correcting for systematic errors, equation (1) is used to estimate the remaining errors.