Question

A gas in a cylinder is held at a constant pressure of 1.80 x 10⁵Pa and is cooled and compressed from 1.70m³ to 1.20 m³. The internal energy of the gas decreases by 1.40 x10⁵J. (a) Find the work done by the gas. (b) Find the absolute value of the heat flow, |Q|, into or out of the gas, and state the direction of the heat flow. (c) Does it matter whether the gas is ideal? Why or why not?

Short answer

(a) If the internal energy of the gas is decreased by$1.40\times {10}^{5}J$ the work done by the gas is$-9\times {10}^4$

(b) The absolute value of the heat flow |Q| is$2.3\times {10}^{5}J$ and the heat flow is negative and the direction of the heat is out from the gas into the surrounding.

(c) No, it does not matter if the gas is ideal or not. As the first law of thermodynamics can be applied on any material ideal or not.

Step-by-step solution

Calculating the work done

Give,

The gas is compressed from$V_1=1.7m^3$ to$V_2=1.2m^3$ and a constant pressure $p=1.8x{10}^5\mathrm{Pa}$, whereas the internal energy is decreased by$4\times {10}^{5}J$

Since the pressure is kept constant, hence the work done is given by:

$\begin{array}{ll}W=p\left(V_2+V_1\right)& \\ =1.8\times {10}^{5}Pa\times \left(1.2m^3-1.7m^3\right)& \\ =-9\times {10}^{4}J& \end{array}$

Therefore, the work done is$-9\times {10}^{4}J$ , since the work done is negative work is done on the gas by the surrounding.

Direction of the heat flow

Now, according to the first law of thermodynamics the heat flow is given by:

$Q=∆U+W$

Since the internal energy is decreased that means the change of energy is negative $(∆U=-1.4\times {10}^{5}J)$and we have also proved that the work done is negative. Now using the first law of thermodynamics to calculate the heat flow through the gas, we have:

$\begin{array}{l}Q=∆U+W\\ =-1.4\times {10}^{5}J+\left(-9\times {10}^{4}J\right)\\ =-2.3\times {10}^{5}J\end{array}$

Now the absolute value of would be

$$\begin{gathered} \left|Q\right|=\left|-2.3\times {10}^{5}J\right| \\ =2.3\times {10}^{5}J \end{gathered}$$

Therefore, the absolute value of heat flow is$2.3\times {10}^{5}J$ and the direction is out from the gas to the surrounding.