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Chapter 17: Problem 50

Estimate the average \(c_{p}\) value, in \(\mathrm{kJ} / \mathrm{kg} \cdot \mathrm{K}\), of a gas if \(522 \mathrm{~kJ} / \mathrm{kg}\) of heat are necessary to raise the temperature from \(300 \mathrm{~K}\) to \(800 \mathrm{~K}\) holding the pressure constant. a) \(1.000\) c) \(1.038\) b) \(1.026\) d) \(1.044\)

Short Answer

Expert verified
Question: Estimate the average specific heat capacity at constant pressure (cₚ) of a gas, given that the heat per unit mass needed to raise its temperature from 300 K to 800 K is 522 kJ/kg. Answer: The average specific heat capacity at constant pressure (cₚ) of the gas is approximately 1.044 kJ/kg·K.

Step by step solution

01

Write down the given conditions and the specific heat capacity formula

We are given that the heat per unit mass needed to raise the temperature from 300 K to 800 K is \(522 \mathrm{~kJ/kg}\). The temperature change is \(800 \mathrm{~K} - 300 \mathrm{~K} = 500 \mathrm{~K}\). We will use the specific heat capacity formula: \(q = mc_{p}\Delta T\)
02

Rearrange the formula to solve for cₚ

We need to find the cₚ value, so we can rearrange the formula to solve for cₚ: \(c_{p} = \frac{q}{m\Delta T}\).
03

Calculate the value of cₚ using the given values

We don't need the mass value in our calculation since we are given the heat per unit mass value. So, we can rewrite the formula accordingly: \(c_{p} = \frac{q}{\Delta T}\). Now, substitute the given values to find cₚ: \(c_{p} = \frac{522 \mathrm{~kJ/kg}}{500 \mathrm{~K}} = 1.044 \mathrm{~kJ/kg \cdot K}\)
04

Compare the calculated value with the given options

The calculated value of cₚ is \(1.044 \mathrm{~kJ/kg \cdot K}\). Comparing this with the given options, we find that the closest option is (d) \(1.044\). So, the average specific heat capacity at constant pressure of the gas is approximately \(1.044 \mathrm{~kJ/kg \cdot K}\).

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