Certainly! Let's solve the problem step by step using the principles of thermodynamics.
### Step 1: Understanding the First Law of Thermodynamics
The first law of thermodynamics can be expressed by the equation:
[tex]\[ \Delta U = Q - W \][/tex]
where:
- [tex]\(\Delta U\)[/tex] represents the change in internal energy,
- [tex]\(Q\)[/tex] represents the heat added to the system,
- [tex]\(W\)[/tex] represents the work done by the system.
### Step 2: Rearrange the Equation
We want to find [tex]\(Q\)[/tex], the heat added. Rearrange the first law of thermodynamics equation to solve for [tex]\(Q\)[/tex]:
[tex]\[ Q = \Delta U + W \][/tex]
### Step 3: Plug in Known Values
From the problem statement, we know:
- The work done by the gas ([tex]\(W\)[/tex]) is 147 J,
- The change in internal energy ([tex]\(\Delta U\)[/tex]) is 306 J.
### Step 4: Substitute the Values into the Equation
Now substitute the given values into the rearranged equation:
[tex]\[ Q = 306\, \text{J} + 147\, \text{J} \][/tex]
### Step 5: Calculate the Heat Added
Perform the addition to find the heat added:
[tex]\[ Q = 306\, \text{J} + 147\, \text{J} = 453\, \text{J} \][/tex]
Thus, the amount of heat added to the system is:
[tex]\[ Q = 453\, \text{J} \][/tex]
