“Thermal Conductivity, Gas Laws & Specific Heat — Complete Guide for Students”

 1.Introduction:

    Heat plays an important role in our daily life and in every branch of science. Understanding how heat flows through materials, how gases behave under different conditions, and how substances absorb heat is essential in physics and engineering.

In this article, we will study:

• Thermal Conductivity
• Temperature Gradient
• Gas Laws
• Ideal Gas Equation
• Heat Capacity and Specific Heat
• Specific Heats of Gases

 2.Thermal Conductivity:

   When one end of a metal rod is heated and the other end is cooled, heat flows continuously from the hot end to the cold end through the process of conduction.

Consider a metal rod:

• One end is kept in a steam bath (hot end).
• The other end is kept in an ice bath (cold end).

   Initially, the temperature of every part of the rod increases. After some time, the temperature at each point becomes constant. This condition is known as the steady state.

At steady state:

• Heat entering from the hot end equals heat leaving from the cold end.
• Temperature decreases gradually from the hot end to the cold end.

Let:

• Temperature at point M = θ₁
• Temperature at point N = θ₂
• Distance between M and N = d

 2.1. Steady State:

  “Steady state is the condition in which the temperature of each point of a conductor remains constant with time, although heat continues to flow through it.”

 2.2.Temperature Gradient:

   “The decrease in temperature per unit distance along the direction of heat flow is called the temperature gradient.”

Temperature gradient = (θ₁ – θ₂ ) / d 

Where:

• θ₁ = temperature at hot point
• θ₂ = temperature at cold point
• d = distance between the two points

 2.3. SI Unit:

kelvin per meter (K/m)

 2.4.Law of Thermal Conductivity:

The amount of heat flowing through a conductor at steady state is:

1. Directly proportional to the cross-sectional area (A)
2. Directly proportional to the temperature gradient
3. Directly proportional to the time of flow (t)

Therefore,      Q α A;       Q α (θ₁ – θ₂ ) / d ;          Q α t

         Q α A.t. (θ₁ – θ₂) / d 

         Q = k .A .t. (θ₁ – θ₂) / d    

Where:

• Q = amount of heat transferred
• A = cross-sectional area
• t = time
• d = distance between points
• k = coefficient of thermal conductivity

 2.5.Coefficient of Thermal Conductivity:

The coefficient of thermal conductivity is defined as:

   “The amount of heat flowing per second through a material of unit cross-sectional area when there is a unit temperature gradient across it.”

 2.6.Units of Thermal Conductivity:

System	Unit
SI Unit	W/m·K
CGS Unit	cal/cm·s·°C
MKS Unit	kcal/m·s·°C

 3. Gas Laws:

   The behavior of gases is described using four physical quantities:

1. Mass
2. Volume
3. Pressure
4. Temperature

 Gas laws study the relationship between pressure, volume, and temperature when one quantity remains constant.

The three important gas laws are:

1. Boyle’s Law
2. Charles’s Law
3. Gay-Lussac’s Law

 3.1. Boyle’s Law:

   PV=constant

  In 1662, Robert Boyle discovered that:

 “At constant temperature, the pressure of a fixed mass of gas is inversely proportional to its volume”.

Thus,   P α 1 / V;   or  PV = constant at constant temperature.

    This indicates that at constant temperature, product of pressure and volume of a fixed mass of gas is constant.

     If a fixed mass of gas at constant temperature T occupying volume V₁ at pressure P₁ undergoes expansion, so that volume changes to V₂ and pressure to P₂, then according to Boyle’s law:

 

P₁ . V₁ = P₂ . V₂ = Constant.

Conclusion:

• If pressure increases, volume decreases.
• If pressure decreases, volume increases.

 3.2. Charles’s Law:

  V/T=constant

    Jacques Charles discovered that:

   “At constant pressure, the volume of a fixed mass of gas is directly proportional to its absolute temperature.”

Thus, V α T ; or  V / T  = constant at constant pressure.

    This indicates that at constant pressure, the ratio of volume of a fixed mass of gas to absolute temperature of gas is constant.

    If a fixed mass of gas at constant pressure P occupying volume V₁ at absolute temperature T₁ undergoes expansion, so that volume changes to V₂ at absolute temperature T₂, then according to Charles’s law :

 V₁ / T₁  = V₂ / T₂ = constant.

Important Relation:

For each degree change in temperature, the volume of sample of a gas changes by the fraction of 1/273.5 of its volume at 0 ^oC.

So   V_t  = V₀ ( 1 + t/ 273 ).

  3.3. Gay-Lussac’s Law:

 P/T=constant

    Gay-Lussac’s law states:

   “At constant volume, the pressure of a fixed mass of gas is directly proportional to its absolute temperature”.

    This indicates that at constant volume, the ratio of pressure of a fixed mass of gas to absolute temperature of gas is constant.

      If a fixed mass of gas at constant volume V, pressure P₁ at absolute temperature T₁ undergoes changes pressure to P₂ at absolute temperature T₂, then according to Gay-Lussac’s law :

 P₁ / T₁  = P₂ / T₂ = constant.

       For each degree change in temperature, the pressure of sample of a gas changes by the fraction of 1/273.5 of its pressure at 0 ^oC.

So   P_t  = P₀ ( 1 + t/ 273 ).

 3.4. Ideal Gas Equation:

It is seen from gas laws for 1 mole of gas:

According to Boyle’s law  P α 1 /V,

According to Gay-Lussac’s law  P α T,

Hence, combining these, we get,  P α T / V,

Or   P V / T = Constant = R  or P V = R T

Where R is Universal gas constant = 8314.9 J / kg. mol. ⁰K .

For n moles of gas  P V = n R T .

The gas which obeys this equation P V = n R T is called Ideal or Perfect gas.

Where:

• P = pressure
• V = volume
• n = number of moles
• R = universal gas constant
• T = absolute temperature

  3.5. Universal Gas Constant:

R = 8.314 J/mol·K
  “   A gas obeying this equation is called an ideal gas.”

 4. Heat Capacity:

Different substances require different amounts of heat to raise their temperatures.

Heat capacity is defined as:

   “The amount of heat required to raise the temperature of a whole body by 1°C.”

 4.1. Specific Heat Capacity:

  “Specific heat capacity is the amount of heat required to raise the temperature of unit mass of a substance by 1°C.”

C = Q / m ∆t

Where:

• Q = heat supplied
• m = mass of substance
• Δt = rise in temperature
• C = specific heat capacity

 4.2. Why Specific Heat is an Intensive Property:

Specific heat is an intensive property because:

• It is defined per unit mass.
• It does not depend on quantity of matter.
• It depends only on the nature of the material.

  4.3. Units:

System	Unit
SI Unit	J/kg·K
CGS Unit	cal/g·°C
MKS Unit	kcal/kg·°C

  4.4. Specific Heat of Gases:

When gases are heated:

• Their volume may increase.
• Their pressure may increase.

Therefore, gases have two specific heats:

1. Specific heat at constant pressure (C_p)
2. Specific heat at constant volume (C_v)

 4.4.1. Specific Heat at Constant Pressure (C_p):

  “Specific heat at constant pressure is the amount of heat required to raise the temperature of 1 kg of gas by 1°C while keeping pressure constant.”

      C_p = Q / m ∆t

 4.4.2. Specific Heat at Constant Volume (C_v):

 “ Specific heat at constant volume is the amount of heat required to raise the temperature of 1 kg of gas by 1°C while keeping volume constant.”

      C_v = Q / m ∆t

4.4.2. Relation between C_p and C_v :

According to Mayer’s relation:

C_p – C_v = R / J

Where:

• R = universal gas constant
• J = Joule’s constant

 4.4.3. Ratio of Specific Heats:

The ratio of specific heats is represented by γ (gamma).

C_p / C_v = γ = 1.4

 5. Conclusion:

Thermal conductivity explains how heat flows through materials, while gas laws describe the behavior of gases under different conditions of pressure, volume, and temperature. Specific heat capacity helps us understand how substances absorb heat energy.

These concepts form the foundation of thermodynamics, heat transfer, and engineering physics.