We can use Graham's law of diffusion to solve this problem, which states that the rate of diffusion of a gas is inversely proportional to the square root of its density.

Let's first find the rate of diffusion of oxygen and chlorine:

Rate of diffusion of oxygen = Volume of oxygen / Time taken = 60 cm^{3} / 10 s = 6 cm^{3}/s

Rate of diffusion of chlorine = Volume of chlorine / Time taken = 100 cm^{3} / 30 s = 3.33 cm^{3}/s

Now, we can set up the following equation using Graham's law:

Rate of diffusion of oxygen / Rate of diffusion of chlorine = sqrt(density of chlorine / density of oxygen)

Plugging in the values we have calculated, we get:

6 cm^{3}/s / 3.33 cm^{3}/s = sqrt(density of chlorine / 1.25)

Simplifying and solving for density of chlorine, we get:

Density of chlorine = (6 cm^{3}/s / 3.33 cm^{3}/s)^{2} * 1.25 = 5.67 g/cm^{3}

Therefore, the density of chlorine is approximately 5.67 g/cm^{3}.