Introduction:
In a galvanic cell, the voltage is determined by the redox reaction occurring at the electrodes. The given reaction involves the metal M, hydrogen ions H+, and the metal ion M* in the half-cell reactions. The voltage of the cell is observed to vary linearly with the log of the square root of the hydrogen pressure and the cube root of the M* concentration. We need to determine the value of x in the reaction.

Explanation:
To understand the relationship between the voltage and the log of the square root of the hydrogen pressure and the cube root of the M* concentration, let's consider the Nernst equation:

Ecell = E°cell - (RT/nF) * ln(Q)

Where:
Ecell is the cell voltage
E°cell is the standard cell potential
R is the gas constant
T is the temperature
n is the number of electrons transferred in the balanced equation
F is Faraday's constant
Q is the reaction quotient

In this case, since the voltage varies linearly with the log of the square root of the hydrogen pressure and the cube root of the M* concentration, we can rewrite the Nernst equation as:

Ecell = E°cell - (RT/nF) * (log(PH2)^(1/2) + log([M*])^(1/3))

We can simplify this equation as:

Ecell = E°cell - (RT/nF) * (1/2 * log(PH2) + 1/3 * log([M*]))

Since the voltage varies linearly with the log of the square root of the hydrogen pressure and the cube root of the M* concentration, the coefficients of log(PH2) and log([M*]) should add up to 1.

1/2 + 1/3 = 5/6

Therefore, the value of x in the reaction M(s) xH* M* X/2) H is 5.

Summary:
The value of x in the reaction M(s) xH* M* X/2) H is 5. This is determined by observing the linear relationship between the voltage of the galvanic cell and the log of the square root of the hydrogen pressure and the cube root of the M* concentration. The coefficients of log(PH2) and log([M*]) in the Nernst equation should add up to 1, which gives us the value of x as 5.

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