Consider the strong electrolytes Z$$_m$$X$$_n$$, U$$_m$$Y$$_p$$ and V$$_m$$X$$_n$$. Limiting molar conductivity ($$\Lambda^0$$) of U$$_m$$Y$$_p$$ and V$$_m$$X$$_n$$ are 250 and 440 S cm$$^2$$ mol$$^{-1}$$, respectively. The value of (m + n + p) is _______.

Given:

Ion	Z$$^{n+}$$	U$$^{p+}$$	V$$^{n+}$$	X$$^{m-}$$	Y$$^{m-}$$
$$\lambda^0$$ (S cm$$^2$$ mol$$^{-1}$$)	50.0	25.0	100.0	80.0	100.0

$$\lambda^0$$ is the limiting molar conductivity of ions

The plot of molar conductivity ($$\Lambda$$) of Z$$_m$$X$$_n$$ vs c$$^{1/2}$$ is given below.

For a strong electrolyte, Kohlrausch’s law of independent ionic migration states that its limiting molar conductivity is the sum of the ionic conductivities multiplied by their stoichiometric coefficients:
$$\Lambda^0=\sum \nu_i\,\lambda_i^0 \qquad -(1)$$

Let the stoichiometric coefficients in the three electrolytes be:

ZmXn    ⟶    $$m\,\text{Z}^{n+}+n\,\text{X}^{m-}$$
UmYp    ⟶    $$m\,\text{U}^{p+}+p\,\text{Y}^{m-}$$
VmXn    ⟶    $$m\,\text{V}^{n+}+n\,\text{X}^{m-}$$

The given ionic conductivities are:

$$\lambda^0_{\text{Z}^{n+}}=50.0,\; \lambda^0_{\text{U}^{p+}}=25.0,\; \lambda^0_{\text{V}^{n+}}=100.0,\; \lambda^0_{\text{X}^{m-}}=80.0,\; \lambda^0_{\text{Y}^{m-}}=100.0\;\;(\text{all in S cm}^2\text{ mol}^{-1}).$$

Electrolyte UmYp
Applying (1):
$$\Lambda^0_{U_mY_p}=m\,\lambda^0_{\text{U}^{p+}}+p\,\lambda^0_{\text{Y}^{m-}}$$ $$250 = 25\,m + 100\,p$$ Divide by 25:
$$m + 4p = 10 \qquad -(2)$$

Electrolyte VmXn
Similarly:
$$\Lambda^0_{V_mX_n}=m\,\lambda^0_{\text{V}^{n+}}+n\,\lambda^0_{\text{X}^{m-}}$$ $$440 = 100\,m + 80\,n$$ Divide by 20:
$$5m + 4n = 22 \qquad -(3)$$

Equations (2) and (3) must be satisfied by positive integers m, n, p.

From (3):
$$5m = 22 - 4n$$ The right-hand side must be a multiple of 5. Testing integer values of $$n$$:

n = 1 → 22 − 4 = 18 (not multiple of 5)
n = 2 → 22 − 8 = 14 (not multiple of 5)
n = 3 → 22 − 12 = 10 (multiple of 5) ⇒ $$m = \frac{10}{5}=2$$

No larger positive value of $$n$$ keeps the right side positive, so the unique solution is
$$m = 2,\; n = 3$$

Substitute $$m = 2$$ into (2):
$$2 + 4p = 10 \;\;\Rightarrow\;\; 4p = 8 \;\;\Rightarrow\;\; p = 2$$

Thus
$$m = 2,\; n = 3,\; p = 2$$

The required sum is
$$(m+n+p) = 2 + 3 + 2 = 7$$

Hence, the value of (m + n + p) is 7.