A gas based geyser heats water flowing at the rate of 5.0 litres per minute from $27^{\circ}\mathrm{C}$ to $87^{\circ}\mathrm{C}$. The rate of consumption of the gas is ____ g/s.

(Take heat of combustion of gas = $5.0 imes 10^4$ J/g) specific heat capacity of water = 4200 J/kg.$^{\circ}$C

Question

A gas based geyser heats water flowing at the rate of 5.0 litres per minute from $27^{\circ}\mathrm{C}$ to $87^{\circ}\mathrm{C}$. The rate of consumption of the gas is ____ g/s.

(Take heat of combustion of gas = $5.0 	imes 10^4$ J/g) specific heat capacity of water = 4200 J/kg.$^{\circ}$C

Accepted Answer

B. **Step 1: Find the heat required per second to warm the water**

The heat energy needed to raise the temperature of a mass is $Q = mc\Delta T$, so the required heating power is:
$$P_{req} = \frac{dQ}{dt} = \frac{dm}{dt}\cdot c\cdot\Delta T$$

Given: water flows at 5.0 litres/min. Since the density of water is 1 kg/L, this is a mass flow rate of:
$$\frac{5.0}{60}	ext{ kg/s}$$

Temperature change: $\Delta T = 87-27 = 60^{\circ}	ext{C}$

Specific heat: $c = 4200	ext{ J/kg·}^{\circ}	ext{C}$

$$P_{req} = \left(\frac{5.0}{60}ight)	imes4200	imes60 = 21000	ext{ J/s}$$

**Step 2: Set up the heat produced by burning gas**

Let $R$ be the rate of gas consumption in g/s. The heat produced per second is:
$$P_{prod} = R	imes H$$

where $H = 5.0	imes10^4	ext{ J/g}$ is the heat of combustion.

$$P_{prod} = R	imes50000	ext{ J/s}$$

**Step 3: Equate heat produced to heat required (assuming 100% efficiency)**

$$R	imes50000 = 21000$$

$$R = \frac{21000}{50000} = 0.42	ext{ g/s}$$

This matches **option B**. This means the geyser must burn gas at a steady rate of 0.42 grams every second to keep up with heating the flowing water continuously.