What is the slope for an isothermal process in PV indicator diagram -

Options

A. `(P)/(V)`
B. `-(P)/(V)`
C. zero
D. `prop`

Answer

To find the slope for an isothermal process in a PV indicator diagram, we can follow these steps:

### Step 1: Understand the Isothermal Process
In an isothermal process, the temperature (T) remains constant. According to the ideal gas law, we have the relationship:

\[ PV = nRT \]

Where:
- \( P \) = pressure
- \( V \) = volume
- \( n \) = number of moles of gas
- \( R \) = universal gas constant
- \( T \) = temperature

Since \( n \), \( R \), and \( T \) are constant for a given gas, we can express this as:

\[ PV = \text{constant} \]

### Step 2: Express Pressure in Terms of Volume
From the equation \( PV = \text{constant} \), we can rearrange it to express pressure \( P \) as:

\[ P = \frac{C}{V} \]

Where \( C \) is a constant.

### Step 3: Differentiate to Find the Slope
To find the slope of the curve in the PV diagram, we need to differentiate \( P \) with respect to \( V \):

1. Differentiate both sides of \( P = \frac{C}{V} \):
- Using the quotient rule or power rule, we get:

\[ \frac{dP}{dV} = -\frac{C}{V^2} \]

### Step 4: Relate the Constant to Pressure and Volume
Since \( C = PV \), we can substitute this back into our equation:

\[ \frac{dP}{dV} = -\frac{PV}{V^2} \]

This simplifies to:

\[ \frac{dP}{dV} = -\frac{P}{V} \]

### Step 5: Interpret the Result
The slope \( \frac{dP}{dV} \) is negative, indicating that as volume increases, pressure decreases, which is characteristic of an isothermal process.

Thus, the slope for an isothermal process in a PV diagram is:

\[ \text{slope} = -\frac{P}{V} \]

### Final Answer
The slope for an isothermal process in a PV indicator diagram is:

\[ \text{slope} = -\frac{P}{V} \]

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