Does farting make the room cooler, or warmer? It's a hot summer day and you've just had some refreshing salad. Unfortunately all the veggies seem to have caused some... gas. Is this good news, or bad news, as far as the physical temperature of the room?
We can treat the fart as a quantity of slightly compressed gas trapped in a rigid, warm vessel. When the gas escapes, its partial pressure will drop while volume increases and therefore it seems like the temperature should slightly drop. On the other hand, the total pressure of the room should slightly increase (if we assume little air flow in/out - bad idea I know). Moreover, as the gas mixes with the room air, perhaps that would result in a more efficient transmission of heat from gas to room air, as opposed to from gas to rigid vessel to room air?
Trying to reason holistically rather than from first principles, it seems like a compressed quantity of gas in a vessel has lower entropy than a dispersed gas. This increase in entropy seems like it would be accompanied by a proportional increase of heat.
Which process prevails? Does the cooling from expansion overpower the warming up from increased pressure and more efficient dispersion? Is the net effect on the room's temperature large, or is it comparatively negligible due to these opposing effects?
For the sake of precision, we can take:

*

*Room temperature as 32C

*Vessel as 37C in thermal equilibrium with the escaping gas

*Room is a cube 4x5x2 m large

*The room is almost airtight but not quite, there is a small gap under the door such that air flux is very small but nonzero

*The air inside the room is efficiently circulated with a fan

 A: The reasoning about the entropy and the gas expansion in the OP is misleading, as it has been pointed out in the comments.
If we add about a mole of molecules with average kinetic energy $\sim k_BT_1$ to a room where molecules have average kinetic energy $\sim k_BT_0$, where $T_1>T_0$, the redistribution of energy will result in some heating. However, the size of the room, as compared to the quantity of gas released, makes this effect negligible for practical purposes. (Even though the temperature inside the human body is typically higher than the skin temperature of 37C, and even the outside temperature of rectum or mouth, measured is routinely for medical perposes, is higher than 37C.)
If the room is aired, the equilibrium with the outside air may establish itself faster than the equilibration of temperatures between the air in the room and the air released. Then there will be no heating at all. Whether this happens can be easily tested experimentally using our sense of smell - if it stays for a while, the airation is not efficient.
A: To answer this question Fart facts will be used, quote:
• On average, a fart is composed of about 59 percent nitrogen, 21 percent hydrogen, 9 percent carbon dioxide, 7 percent methane and 4 percent oxygen. Less than 1 percent of their makeup is what makes farts stink.
• The temperature of a fart at time of creation is 98.6 degrees Fahrenheit.
• Farts have been clocked at a speed of 10 feet per second.
• A person produces about half a litre of farts a day.
• Women fart as much as men.
The most useful of the above 'facts' are that the fart is at about 98.6F or 37F upon emission and that it's mass would be 0.5g, assuming the same density as air (1kg per cubic metre) and that the fart was large, a days quota.
Then we have the room's mass of air, $40m^3$, $40kg$ at 32C, mixing with $0.0005kg$ of fart at 37C.
$$40.0005T = 40\times 32 +    0.0005\times 37$$
$$T= 32.00006C$$
An insignificant change.
This question brings Billy Connolly's joke to mind.
"Apparently the average person farts 6 times per day...but how do they get the figures?"
