Question

can two isothermal curves intersect, if no, why

Answer 1

No.

I am answering this question considering that you are talking about ideal gases undergoing isothermal process

For ideal gas, we know that

PV=nRT

If the process is isothermal, change in temperature is 0 and hence T becomes constant. Also R and n are already constants. Hence the entire RHS term becomes a constant

So you get a graph of

PV=K, where K is some constant.

So you get a graph of a rectangular hyperbola in the first quadrant. We don't take the graph of third quadrant as neither pressure nor volume can be -ve

Now as temperature increases, the value of K increase and the graph shifts a bit towards +ve x and y axis. So the graphs never meet each other .

You can also think about it with mathematics. If two graphs xy=a and xy=b are two graphs with a and b as two non zero constants not equal to each other, they can never intersect each other as the point of intersection would satisfy bother the graphs but product of two numbers(co-ordinates of the point of intersection) cannot give 2 different answers (a and b). Hence the graphs never intersect