The total entropy change `(DeltaS_("total"))` for the system and surrounding of a spontaneous process is given by

Options

A. `DeltaS_("total")=DeltaS_("system")+DeltaS_("surr")gt0`
B. `DeltaS_("total")=DeltaS_("system")+DeltaS_("surr")lt0`
C. `DeltaS_("system")=DeltaS_("total")+DeltaS_("surr") gt 0`
D. `DeltaS_("surr")=DeltaS_("total")+DeltaS_("system")lt0`

Explanation

To determine the total entropy change \((\Delta S_{\text{total}})\) for a spontaneous process, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Entropy Change**:
The total entropy change for a process is given by the sum of the entropy change of the system and the surroundings. Mathematically, this can be expressed as:
\[ \Delta S_{\text{total}} = \Delta S_{\text{system}} + \Delta S_{\text{surroundings}} \]

2. **Condition for Spontaneity**:
For a process to be spontaneous, the total entropy change must be positive:
\[ \Delta S_{\text{total}} > 0 \]

3. **Entropy Change of the System**:
The entropy change of the system can be related to the surroundings. If we denote the entropy change of the surroundings as \(\Delta S_{\text{surroundings}}\), we can express it as:
\[ \Delta S_{\text{surroundings}} = -\frac{q_{\text{rev}}}{T} \]
where \(q_{\text{rev}}\) is the heat exchanged reversibly and \(T\) is the temperature.

4. **Combining the Equations**:
Substituting the expression for \(\Delta S_{\text{surroundings}}\) into the equation for \(\Delta S_{\text{total}}\):
\[ \Delta S_{\text{total}} = \Delta S_{\text{system}} - \frac{q_{\text{rev}}}{T} \]

5. **Conclusion**:
For a spontaneous process, since \(\Delta S_{\text{total}} > 0\), it implies:
\[ \Delta S_{\text{system}} + \left(-\frac{q_{\text{rev}}}{T}\right) > 0 \]
This means that the entropy change of the system must be sufficiently large to overcome any decrease in the entropy of the surroundings due to heat loss.

### Final Answer:
The total entropy change \((\Delta S_{\text{total}})\) for a spontaneous process is positive:
\[ \Delta S_{\text{total}} > 0 \]