Question

A ${\mathbf{5}}{\text{ }}{\mathbf{mol}}$ of gas at $5{\text{ }}atm$ pressure contained in a $100L$ cylinder absorbed $30.26{\text{ }}kJ$ of heat when it expanded to $200L$ at $2atm$ pressure. The change in internal energy of gas is:

Answer 1

Gases are known to be complicated. They are full of energetic gas molecules. These molecules collide and make sure to interact with each other. It's not very easy to describe a real gas, the concept of ideal gas was created. This concept helps us understand the properties of gas in different conditions.

Complete step-by-step answer: Internal energy is the motion of molecules is disordered; this energy is then associated. It is the invisible microscopic energy on both atomic and molecular scale. When we talk about an ideal monatomic gas we must know that the translational kinetic energy of linear motion of atoms can be well described by kinetic theory. In polyatomic gases, there is rotational and vibrational kinetic energy present as well. Internal energy is related to intermolecular attractive forces.
There are no gases that are exactly ideal, but there are plenty of gases that are close enough that the concept of an ideal gas is an extremely useful approximation for many situations. If the pressure of the gas is too large (e.g., hundreds of times larger than atmospheric pressure), or the temperature is too low there can be significant deviations from the ideal gas law. The ideal gas law is the simple formula which relates pressure, volume, and temperature of an ideal gas.
The relationship can be seen by the equation:
$PV = nRT$
Now, change in internal energy,
$\Delta u = q - \Delta p\Delta v$
$\;\; = 30.26 - [(2 - 5)(200 - 100)]/1000$
$= 30.56kJ$

Note: Where $P$ is the pressure of the gas, $V$ is the volume taken up by the gas, $T$ is the temperature of the gas, $R$ is the gas constant, and $n$ is the number of moles of the gas. There is another way to write this equation. In this, if the number of moles of gas doesn’t change, then the quantity $nR$and $N{k_b}$ are constant for a gas. Thus by moving the values pressure, volume and temperature onto the same side of ideal gas law we get, $nR = N{K_b}$ $= PV/T =$ constant.
This means that if the number of moles of gas are the same then the quantity $PV/T$is constant irrespective of the process through which the gas is taken.
Thus we can say that,
${P_1}{V_1}/{T_1} = {P_2}{V_2}/{T_2}$