A metal has a work function `phi_(0)` and its corresponding threshold wavelength is `lambda_(0)`. If the threshold wavelength for a metal whose work function is `(phi_(0))/(3)` is `n lambda_(0)`, then what is the value of n ?

Answer

To solve the problem, we need to relate the work function of the metal to its threshold wavelength using the equation that connects them. The work function (φ) is given by the equation:

\[ \phi = \frac{hc}{\lambda} \]

where:
- \( h \) is Planck's constant,
- \( c \) is the speed of light,
- \( \lambda \) is the wavelength.

### Step 1: Establish the relationship for the first metal

For the first metal, we have:

\[ \phi_0 = \frac{hc}{\lambda_0} \]

This equation shows that the work function \( \phi_0 \) corresponds to the threshold wavelength \( \lambda_0 \).

### Step 2: Establish the relationship for the second metal

For the second metal, whose work function is \( \frac{\phi_0}{3} \), we can write:

\[ \frac{\phi_0}{3} = \frac{hc}{n\lambda_0} \]

This equation indicates that the work function of the second metal is equal to \( \frac{hc}{n\lambda_0} \), where \( n\lambda_0 \) is the new threshold wavelength.

### Step 3: Substitute the first equation into the second

Now, we substitute the expression for \( \phi_0 \) from the first equation into the second equation:

\[ \frac{1}{3} \left(\frac{hc}{\lambda_0}\right) = \frac{hc}{n\lambda_0} \]

### Step 4: Simplify the equation

We can simplify this equation by canceling \( hc \) and \( \lambda_0 \) from both sides:

\[ \frac{1}{3} = \frac{1}{n} \]

### Step 5: Solve for \( n \)

To find \( n \), we can take the reciprocal of both sides:

\[ n = 3 \]

### Conclusion

Thus, the value of \( n \) is:

\[ \boxed{3} \]

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