Question

A gas is allowed to expand in a well-insulated container against constant external pressure of \[2.5atm\]from an initial volume of \[2.50L\] to a final volume of \[4.50L\]. The change in internal energy \[\Delta U\] of the gas in joules will be:
A \[+505J\]
B \[+1136.25J\]
C \[-500J\]
D \[-505J\]

Answer 1

Hint: The internal energy of a system can be determined by calculating the simplest possible system: an ideal gas. Because the particles in an ideal gas do not interact, this system has no potential energy. The internal energy of an ideal gas is therefore the total sum of the kinetic energies of the particles (molecules or atoms) in the gas.

Complete step by step answer:
We have known that the change in internal energy can be given by the following expression,
\[\Delta U=q+W\]
\[\Delta U= \] Change in internal energy of the system
\[q= \] Heat transfer between the system
\[W= \] Work done by the system
According to the question, it is given that the gas is allowed to expand in a well-insulated container. So, the heat transfer is equal to zero. Hence, \[q=0\]

Hence, we can write the change in internal energy of the system is equal to work done by the gas only. i.e,
\[\Delta U=W=-P\Delta V\]
\[P= \]Pressure of the gas
\[\Delta V= \]Change in volume of the gas
In the question we have given that;
[P=2.5atm
\[\Delta V=(4.50L-2.50L)=2L\]
Substitute these values in the equation. Hence we get as
\[\Rightarrow \Delta U=W=-P\Delta V=-2.5atm\times 2L\]
\[\Rightarrow \Delta U=-5.0Latm\]
The change in internal energy is equal to \[-5.0Latm\]. We have to convert this value into joules. So, we have to multiply this value by \[\dfrac{101.33J}{1Latm}\]
\[\Rightarrow \Delta U=-5.0Latm\times \dfrac{101.33J}{1Latm}=-505J\]
Here we got the change internal energy is equal to \[-505J\]

So, the correct answer is Option D.

Note: According to first law of thermodynamics, In a closed system (i.e. there is no transfer of matter into or out of the system), the change in internal energy of the system (ΔU) is equal to the difference between the heat supplied to the system (q) and the work (W) done by the system on its surroundings.