The relation between `DeltaU` is :

Options

A. `DeltaH = Delta U + RT`
B. `DeltaH = Delta U - Delta_(g)nRT`
C. `DeltaH = Delta U + Delta_(g)nRT`
D. `DeltaU = Delta H + Delta_(g)nRT`

Explanation

To derive the relationship between the change in internal energy (ΔU) and the change in enthalpy (ΔH), we can follow these steps:

### Step 1: Start with the definition of enthalpy
The enthalpy (H) is defined as:
\[ H = U + PV \]
where \( U \) is the internal energy, \( P \) is the pressure, and \( V \) is the volume.

### Step 2: Differentiate the enthalpy
Taking the differential of both sides, we have:
\[ dH = dU + PdV + VdP \]

### Step 3: Consider constant pressure conditions
Under constant pressure conditions (which is often the case in chemical reactions), \( dP = 0 \). Therefore, the equation simplifies to:
\[ dH = dU + PdV \]

### Step 4: Relate changes in volume to the number of moles
For reactions involving gases, we can express the change in volume in terms of the change in the number of moles of gas. If \( n_1 \) is the number of moles of gaseous reactants and \( n_2 \) is the number of moles of gaseous products, then:
\[ \Delta n = n_2 - n_1 \]

### Step 5: Use the ideal gas law
According to the ideal gas law, we have:
\[ PV = nRT \]
Thus, for the reactants and products, we can write:
- For reactants: \( PV_1 = n_1RT \)
- For products: \( PV_2 = n_2RT \)

### Step 6: Substitute into the enthalpy equation
Substituting \( PdV \) (where \( dV = V_2 - V_1 \)) into the equation gives:
\[ dH = dU + P(V_2 - V_1) \]
Using the ideal gas law, we can express this as:
\[ dH = dU + RT(n_2 - n_1) \]

### Step 7: Final relationship
Thus, we arrive at the relationship:
\[ \Delta H = \Delta U + \Delta n RT \]
where \( \Delta n = n_2 - n_1 \).

### Conclusion
The final relationship between the change in enthalpy and the change in internal energy is:
\[ \Delta H = \Delta U + \Delta n RT \]

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