At 298 K, the limiting molar conductivity of a weak monobasic acid is 4 $$\times$$ 10² S cm² mol^$$-$$1. At 298 K, for an aqueous solution of the acid the degree of dissociation is $$\alpha$$ and the molar conductivity is y $$\times$$ 10² S cm² mol^$$-$$1. At 298 K, upon 20 times dilution with water, the molar conductivity of the solution becomes 3y $$\times$$ 10² S cm² mol^$$-$$1.

The value of y is __________.

Consider a 70% efficient hydrogen-oxygen fuel cell working under standard conditions at 1 bar and 298 K. Its cell reaction is

$${H_2}(g) + {1 \over 2}{O_2}(g)\buildrel {} \over \longrightarrow {H_2}O(l)$$

The work derived from the cell on the consumption of 1.0 $$ \times $$ 10^$$-$$3 mole of H₂(g) is used to compress 1.00 mole of a monoatomic ideal gas in a thermally insulated container. What is the change in the temperature (in K) of the ideal gas?

The standard reduction potentials for the two half-cells are given below :

$${O_2}(g) + 4{H^ + }(aq) + 4{e^ - }\buildrel {} \over \longrightarrow 2{H_2}O(l),$$

$${E^o} = 1.23V$$

$$2{H^ + }(aq) + 2{e^ - }\buildrel {} \over \longrightarrow {H_2}(g),$$

$${E^o} = 0.00\,V$$

Use, $$F = 96500\,C\,mo{l^{ - 1}}$$, $$R = 8.314\,J\,mo{l^{ - 1}}\,{K^{ - 1}}$$.

Consider an electrochemical cell :

$$A\left( s \right)\left| {{A^{n + }}\left( {aq,2M} \right)} \right|{B^{2n + }}\left( {aq,1M} \right)\left| {B\left( s \right).} \right.$$

The value of $$\Delta {H^ \circ }$$ for the cell reaction is twice that of $$\Delta {G^ \circ }$$ at $$300$$ $$K.$$ If the $$emf$$ of the cell is zero, the $$\Delta {S^ \circ }$$ (in $$J\,{K^{ - 1}}mo{l^{ - 1}}$$) of the cell reaction per mole of $$B$$ formed at $$300$$ $$K$$ is ___________.

(Given: $$\ln \left( 2 \right) = 0.7,R$$ (universal gas constant) $$ = 8.3J\,{K^{ - 1}}\,mo{l^{ - 1}}.$$ $$H,S$$ and $$G$$ are enthalpy, entropy and Gibbs energy, respectively.)

For the electrochemical cell,

$$\left. {Mg\left( s \right)} \right|M{g^{2 + }}\left( {aq,1\,M} \right)\left\| {C{u^{2 + }}} \right.\left( {aq,1M} \right)\left| {Cu\left( s \right)} \right.$$

the standard $$emf$$ of the cell is $$2.70$$ $$V$$ at $$300$$ $$K.$$ When the concentration of $$M{g^{2 + }}$$ is changed to $$x$$ $$M,$$ the cell potential changes to $$2.67$$ $$V$$ at $$300$$ $$K.$$ The value of $$x$$ is ___________.

(given, $${F \over R} = 11500\,K{V^{ - 1}},$$ where $$F$$ is the Faraday constant and $$R$$ is the gas constant, In $$(10=2.30)$$