The index of refraction of a material is equal to , where  is the speed of light in a vacuum and  is the speed of light in the material. In the material given, the light travels a distance of  (since it travels to the mirror and back) in a time . Speed is distance over time, so the light's speed in the material is . Thus the index of refraction is 

Which when simplified, gives:

Using Snell's law:

Using Snell's law:

This is a direct application of snell's law. 

This gives

Using Snell's law:

To find the time, we need to use  where  is the distance traveled. Putting in our values, we get:

Using Snell's law:

 . The  subscript stands for incident, while the  stands for refraction. Here, we want to solve for the refraction angle. Let's plug in numbers:

Snell's Law states:

. The  subscript stands for incident, while the  stands for refraction. Here, we want to solve for the refraction angle. Let's plug in numbers:

We can use Snell's law:

At the first instance of total internal reflection, the angle of light on the air side will be  to the vertical, so the equation becomes:

Rearranging for the index of refraction of water, we get:

What this question is essentially asking for is the critical angle, which is the angle at which total internal reflection occurs. When this happens, the light will bend at an angle of  from the normal, or essentially along the interface of the two media. To solve for this value, we'll need to make use of Snell's law:

Where  is the index of refraction for benzene and  is the index of refraction for water.

To determine the critical angle, we'll make the following adjustments:

And plugging in values:

This is the angle where the incident light ray will be refracted at exactly . Any incident angle larger than this value will result in total internal reflection, where the light ray will essentially bounce back into the medium in which it originated.

Also, one very important thing to keep in mind is that total internal reflection can only occur when light travels from a medium with a higher index of refraction to a second medium with a lower index of refraction. So in the example given, total internal reflection can only occur when light travels from benzene (higher index) to water (lower index) and not the other way around.

The equation to find total internal reflection is a special case of Snell's Law, with the refraction angle set to . This new equation looks like this:

To find the critical angle, we can rearrange the equation.

As you can tell from this, the index of refraction of the medium the light is traveling from must be higher than the index of refraction of the other medium. Our numbers qualify. Now, we can plug in our values to find the critical angle.