The isothermal bulk modulus of elasticity of a gas is 1.5 * 100000 N/m^2. What will be its adiabatic bulk modulus of elasticity?
isothermal bulk modulus of elasticity K(T) = P
adiabatic bulk modulus of elasticity K(a) = γP,
where P is the pressure, γ is ratio of specific heats – adiabatic exponent ( is the ratio of the heat capacity at constant pressure (C(p)/) to heat capacity at constant volume (C(v) )
γ = C(p)/ C(v)
The heat capacity ratio ( γ) for an ideal gas can be related to the degrees of freedom of a molecule, therefore, it is necessary to know how many atoms are in the gas molecule
To find the adiabatic bulk modulus of elasticity (K), you need to know the isothermal bulk modulus of elasticity (K_isothermal) and the specific heat ratio of the gas (γ).
The relationship between the isothermal bulk modulus (K_isothermal) and the adiabatic bulk modulus (K) is given by the formula:
K = K_isothermal * (γ / (γ - 1))
In this case, the isothermal bulk modulus of elasticity (K_isothermal) is given as 1.5 * 100000 N/m^2. But we need to determine the specific heat ratio (γ) of the gas.
The specific heat ratio (γ) depends on the type of gas. For example, for an ideal monoatomic gas, the specific heat ratio is 5/3 (γ = 5/3). For a diatomic gas, such as most gases in the atmosphere, the specific heat ratio is approximately 7/5 (γ = 7/5).
Once you know the specific heat ratio (γ) for the gas, you can substitute it into the formula to find the adiabatic bulk modulus (K).
For instance, let's assume we are dealing with a diatomic gas with γ = 7/5. We can now calculate the adiabatic bulk modulus as follows:
K = (1.5 * 100000 N/m^2) * (7/5) / ((7/5) - 1)
Simplifying further:
K = (1.5 * 100000 N/m^2) * (7/5) / (2/5)
K = (1.5 * 100000 N/m^2) * (7/5) * (5/2)
K = (1.5 * 100000 N/m^2) * (7/2)
K ≈ 525000 N/m^2
Therefore, the adiabatic bulk modulus of elasticity of the gas would be approximately 525000 N/m^2.