To solve the problem of finding the slopes between \( PV \) versus \( V \) and \( PV \) versus \( P \) at constant temperature, we can follow these steps:
### Step 1: Understand the Ideal Gas Law
The ideal gas law is given by the equation:
\[
PV = nRT
\]
where:
- \( P \) = pressure
- \( V \) = volume
- \( n \) = number of moles of gas
- \( R \) = universal gas constant
- \( T \) = temperature
At constant temperature, \( nRT \) is a constant.
### Step 2: Analyze the First Slope (PV vs. V)
To find the slope of the graph of \( PV \) versus \( V \):
- Since \( PV = nRT \) is constant, we can express this as:
\[
PV = C \quad (where \, C = nRT)
\]
- Rearranging gives:
\[
P = \frac{C}{V}
\]
This indicates that as \( V \) increases, \( P \) decreases, and the relationship is hyperbolic.
However, if we consider \( PV \) as a function of \( V \):
- The value of \( PV \) remains constant regardless of changes in \( V \).
- Therefore, when plotting \( PV \) against \( V \), the graph will be a horizontal line, indicating that the slope is 0.
### Step 3: Analyze the Second Slope (PV vs. P)
Now, we find the slope of the graph of \( PV \) versus \( P \):
- From the ideal gas equation, we can express \( V \) as:
\[
V = \frac{C}{P}
\]
- This shows that as \( P \) increases, \( V \) decreases, again indicating a hyperbolic relationship.
However, if we consider \( PV \) as a function of \( P \):
- The value of \( PV \) remains constant regardless of changes in \( P \).
- Therefore, when plotting \( PV \) against \( P \), the graph will also be a horizontal line, indicating that the slope is again 0.
### Conclusion
Thus, the slopes between \( PV \) versus \( V \) and \( PV \) versus \( P \) at constant temperature are both 0. Therefore, the answer is:
\[
\text{0 and 0}
\]
### Final Answer
The correct option is **0 and 0**.
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