 ## Heat Equation and Specific Heat Capacity

Subject: Science

#### Overview

Heat equation is Q = msdt where, S is proportionality constant and is known as specific heat capacity. This note details about heat equation and specific heat capacity along with principle of calorimetry.

### Heat equation:

Let us consider a body having mass 'm' is heated so that its temperature changes from t1 to t2. Therefore, a change in temperature is t2 - t1. The amount of heat "Q" depends on two factors i.e. mass of the body and temperature differences

I.e. Q∝m ........(i)

Q∝( t2 - t1)

Q∝dt ............ (ii) Where dt is temperature difference

Combining equation (i) and (ii)

Q∝mdt

Q = smdt

Therefore, Q = msdt

Where, 's' is proportionality constant and is known as specific heat capacity.

### Specific Heat capacity:

we have,

Q = msdt

or, s = $\frac{Q}{mdt}$

When,

m = 1 kg

dt = 1 degree Celsius

s = Q

Thus, Specific heat capacity is defined as the amount of heat required to change the temperature of 1 kg of substances through 1°C or 1k

### Unit of specific heat capacity:

We have,  S =$\frac{Q}{mdt}$

SI unit of heat energy (Q) is joule

SI unit of mass (m) is kilogram

SI unit of change in temperature (dt) is Kelvin

Therefore, S =$\frac{J}{kgK}$=Jkg-1K-1

Note: Different substances have different specific heat capacity. The substance that has more specific heat capacity changes its temperature slowly (when it is heated its temperature rise up slowly and when it is cooled its temperature fall down slowly). The substances that have less specific heat capacity changes its temperature fast (when it is heated its temperature rise up quickly)

#### Specific heat capacity of some materials:

 Materials Specific heat capacity Gold 134 Jkg-1K-1 Mercury 138 Jkg-1K-1 Sand 800 Jkg-1K-1 Ice 2100 Jkg-1K-1 Water 4200 Jkg-1K-1

### Thermal capacity is defined as the amount of heat required to change the temperature of the substance through 1 degree Celsius. It is denoted by "C".Thermal capacity:

Mathematically,

C = m s where,

C = thermal capacity

M = mass of substance

S = specific heat capacity.

Its SI unit is J/K or J/°C

#### Principal of calorimetry:

As we know that, heat transfers from the hotter body to a cooler body when the two bodies are kept in contact with each other. According to the principle of conservation of energy, energy can neither be created nor be destroyed. Thus, the heat energy lost from the body kept at a higher temperature is equal to the heat energy gained by the body at the lower temperature. There is an assumption that there is no loss of heat in the surrounding, so there will be an exchange of heat between the hot and cold bodies. The exchange of heat continues until they obtain the same temperature. It is said to be principle of calorimetry.

It states that "The amount of heat lost by the hotter body is equal to the amount of heat gained by the colder body, avoiding external loss of heat."

Mathematically,

Heat lost by a body at higher temperature = Heat gained by a body at a lower temperature

i.e. m1 × s1 × dt1 = m2 × s2 × dt2

where, m1 = mass of the body (1)

m2 = mass of the body (2)

s1 = specific heat capacity (1)

s2 = specific heat capacity (2)

dt1 = change in temperature (1)

dt2 = change in temperature (2)

Let us consider two bodies A and B where the masses are m1 and m2. Let, t1 and t2 be the initial temperature of A and B respectively. Let specific heat capacity of A and B be S1 and S2 respectively. If A is hotter body and B is the older body when they are kept in thermal contact, A will lose heat and B will gain heat until their temperature becomes equal "t".

Hence,

Amount of heat gained by B (Q2) = m2S2(t - t2)

Amount of heat lost by A (Q1) = m1S1(t1-t)

According to principle of calorimetry amount of heat loosed by A is equal to the amount of heat gained by B

Therefore, m1 S1(t1-t) = m2 S2 (t-t2)

##### Things to remember
1. Heat equation is Q = msdt where S is proportionality constant and is known as specific heat capacity.
2. Specific heat capacity is defined as the amount of heat required to change the temperature of 1 kg of substances through 1°C or 1k.
3. The substance that has more specific heat capacity changes its temperature slowly (when it is heated its temperature rise up slowly and when it is cooled its temperature fall down slowly).
4. The amount of heat gained by the body at a lower temperature is equal to the heat lost by the body at a higher temperature; this relation is said to be principle of calorimetry.
5. The thermal capacity of a body is defined as the amount of heat required to raise the temperature of the body by 1⁰C i.e. C = m × s.
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##### Videos for Heat Equation and Specific Heat Capacity
It is the amount of heat energy required to raise the temperature of 1 gm of water by 1 degree Celsius.
We can define temperature in two different ways:
1. The degree of hotness and coldness of a substance is called temperature.
2. The intensity of molecular vibration of the substance is called its temperature.
 Heat Temperature It is the sum of kinetic energy possessed by the substance. It is the intensity of kinetic energy of the molecules of the substance. Heat is the form of energy that gives us sensation of warmth. The degree of hotness and coldness of a substance is called temperature. It is measured in Joules. It is measured in degree Celsius. It is the cause. It is an effect.
The heat transfers from hot substance to the cold substance. This is because hot bodies have high kinetic energy than cold bodies.

The specific heat capacity of water is 4200 J/ Kg 0C means that 4200 J heat energy is required to increase or decrease the temperature of 1 kg water by 1 0C.

Given,

Heat amount (Q) =1000J

Mass of iron (m) = 2 kg

Temperature decreased (t) = 90°C -15°C

= 75°C

Specific heat capacity (s)=?

We know,

Q = m × s ×t

or, s = $\frac{Q}{mt}$

= $\frac{1000}{2× 75}$

= 6.67J/kg°c

Specific heat capacity of a substance can be defined as the amount of heat required to raise the temperature of kg of the substance by 1°c.The specific heat of the substance is generally represented by letters.The unit of specific heat capacity is J/kg°c.

Given,

Mass of the iron (m) =150 gm =0.150kg.

Increase in temperature (t) = 150°c - 25°c =125°c

The specific heat of the iron (s) = 480J/kg°c

Quantity of heat required (Q) = ?

We have from the relation

Q = m× s× t

= 0.150× 480× 125

= 9,000 Joules

Hence , 9000 Joules heat is required.