Problem 1

The “clathrate gun” scenario: ocean-floor methane hydrates could decompose explosively as oceans warm. Upon decomposition of 1.00 g of a methane hydrate of fixed composition at 25 °C and 101.3 kPa, 205 cm

^3

of methane is released. (a) Determine

n

\ce{CH4 \cdot n H2O}

. (b) Real hydrate is close to

\ce{CH4 \cdot 6 H2O}

, stable at 1 atm down to its decomposition at

-81

°C; write its decomposition to methane and ice,

\Delta H = +17.47

kJ mol

^{-1}

. Assuming

\Delta H

is T- and p-independent, that the volume change equals the released methane volume, and that methane is ideal, find the external pressure at which decomposition occurs at

-5

°C. (c) What is the minimum possible depth of pure liquid water at which methane hydrates are stable? First choose the minimum temperature at which the hydrate can coexist with liquid water: 272.9 K, 273.15 K, or 273.4 K. (d) In Lake Baikal, hydrate samples raised from 1400 m began to decompose at 372 m depth (

\Delta_{fus}H

(ice) = 6.01 kJ mol

^{-1}

); find the water temperature at 372 m. (e) Total methane in hydrates is

\geq 5 \times 10^{11}

t; by how many degrees would Earth's atmosphere heat if it all burned (

\Delta_c H(\ce{CH4}) = -889

kJ mol

^{-1}

, atmospheric heat capacity

4 \times 10^{21}

J K

^{-1}

Step 5 of 5: Atmospheric heating

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Q = \dfrac{5 \times 10^{11} \times 10^{6}}{0.016} \times 889 \times 10^{3} = 2.78 \times 10^{22}\ \text{J};\quad \Delta T = \dfrac{2.78 \times 10^{22}}{4 \times 10^{21}} \approx 7\ \text{K}

Analysis

Burning $5 \times 10^{11}$ t of methane releases $2.78 \times 10^{22}$ J; spread over the atmosphere's heat capacity of $4 \times 10^{21}$ J K $^{-1}$ this is about 7 K of warming.