Which one of the following relationships when graphed does not give a straight line for helium gas?
I. K.E. and T at constant pressure and volume
II. P v/s V at constant temperature for a constant mass
III. V v/s 1/T at constant pressure for a constant mass

Options

A. II
B. II and III
C. III
D. I

Explanation

To solve the question, we need to analyze each of the given relationships and determine whether they yield a straight line when graphed for helium gas.

### Step-by-Step Solution:

1. **Analyze Relationship I: K.E. and T at Constant Pressure and Volume**
- The kinetic energy (K.E.) of a gas is given by the formula:
\[ K.E. = \frac{3}{2} nRT \]
- Here, \(n\) (number of moles) and \(R\) (universal gas constant) are constants. Therefore, K.E. is directly proportional to temperature \(T\) when pressure and volume are held constant.
- When graphed, this relationship (K.E. vs. T) will yield a straight line since it is a linear relationship.

**Hint:** Remember that direct proportionality between two variables results in a straight line graph.

2. **Analyze Relationship II: P vs. V at Constant Temperature for Constant Mass**
- According to Boyle's Law, for a given mass of gas at constant temperature, the product of pressure (P) and volume (V) is constant:
\[ PV = k \quad (k \text{ is a constant}) \]
- When we rearrange this equation to express pressure as a function of volume:
\[ P = \frac{k}{V} \]
- This relationship describes a hyperbola when graphed (P vs. V), indicating that it does not yield a straight line.

**Hint:** Recall that inverse relationships typically result in hyperbolic graphs.

3. **Analyze Relationship III: V vs. 1/T at Constant Pressure for Constant Mass**
- We can express the relationship using the ideal gas law:
\[ PV = nRT \implies V = \frac{nRT}{P} \]
- If we consider \(V\) as a function of \(1/T\) at constant pressure, we can rearrange it:
\[ V = k \cdot \frac{1}{T} \quad (k = \frac{nR}{P}) \]
- This indicates that volume \(V\) is inversely proportional to temperature \(T\). When graphed, this will also yield a hyperbolic curve.

**Hint:** Inverse relationships can also lead to hyperbolic graphs, similar to the second relationship.

### Conclusion:
From the analysis:
- Relationship I gives a straight line.
- Relationships II and III do not give straight lines; they yield hyperbolas.

Thus, the correct answer is that the relationships that do not give a straight line for helium gas are **II and III**. The question specifically asks for the one that does not yield a straight line, so the answer is **II**.

### Final Answer:
**II. P vs. V at constant temperature for a constant mass does not give a straight line.**