Question

At very low temperatures the molar heat capacity of rock salt varies with temperature according to Debye's T^3 law: C=kT^3/ θ^3 where k = 1940J/(mol⋅K) and θ=281K.Required:a. How much heat is required to raise the temperature of 1.50 mol of rock salt from 10.0 K to 40.0 K? b. What is the average molar heat capacity in this range? c. What is the true molar heat capacity at 40.0 K?

Answer 1

At very low temperatures, the molar heat capacity of rock salt can be calculated using Debye's T^3 law. To find the heat required to raise the temperature of rock salt from 10.0 K to 40.0 K, we can use the formula Q = nCΔT. The average molar heat capacity and true molar heat capacity at 40.0 K can also be determined using the given equation and values.

Step-by-step explanation:

The molar heat capacity of rock salt at very low temperatures can be calculated using Debye's T^3 law. Debye's law states that C = kT^3/ θ^3, where C is the molar heat capacity, k is a constant (1940 J/(mol⋅K)), T is the temperature, and θ is a characteristic temperature.

a. To calculate the heat required to raise the temperature of 1.50 mol of rock salt from 10.0 K to 40.0 K, we can use the formula Q = nCΔT, where Q is the heat, n is the number of moles, C is the molar heat capacity, and ΔT is the change in temperature. Plugging in the values, we get Q = (1.50 mol)(1940 J/(mol⋅K))(40.0 K - 10.0 K) = 232,800 J. So, 232,800 J of heat is required.

b. The average molar heat capacity in this temperature range can be calculated by taking the average of the molar heat capacities at the initial and final temperatures. So, (C1 + C2)/2 = [(1940 J/(mol⋅K))(10.0 K)^3/ θ^3 + (1940 J/(mol⋅K))(40.0 K)^3/ θ^3]/2. Substitute the value of θ (281 K) to calculate the average molar heat capacity.

c. To find the true molar heat capacity at 40.0 K, we can substitute the value of T into the equation C = kT^3/ θ^3. So, C = (1940 J/(mol⋅K))(40.0 K)^3/ θ^3. Substitute the value of θ (281 K) to calculate the true molar heat capacity at 40.0 K.

Answer 2

The heat required to raise the temperature of 1.50 mol of rock salt from 10.0 K to 40.0 K can be calculated using Debye's T^3 law by integrating the heat capacity function, and the average and true molar heat capacities can be determined similarly through integration and substitution of the known values into the formula.

Step-by-step explanation:

To calculate the heat required to raise the temperature of 1.50 mol of rock salt from 10.0 K to 40.0 K using Debye's T3 law, we integrate the heat capacity function over the given temperature range. The equation C = (kT3) / θ3 is used where k = 1940 J/(mol·K) and θ = 281 K.

For part (a), the heat required (Q) is given by:

Q = ∫10K40K C(T) dT

The average molar heat capacity (Cavg) for part (b) is calculated as:

Cavg = (1/ΔT) ∫T1T2 C(T) dT

For part (c), the true molar heat capacity at 40.0 K is simply:

C(40K) = (k * 40K3) / θ3

By solving these integrals and calculations, the student can obtain the required heat to raise the temperature, and the molar heat capacities.