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Physics

Heat

Easy

Question

Find the amount of work done to increase the temperature of one mole of an ideal gas by 30°C , if it is expanding under condition $V proportional to T to the power of 2 divided by 3 end exponent$ . R is gas constant

32R
15R
20R
25R

The correct answer is: 20R

$text idea gas equation end text P V equals n R T text For one mole of ideal gas end text P V equals R T not stretchy rightwards double arrow P equals fraction numerator R T over denominator V end fraction text Given that end text V proportional to T to the power of 2 divided by 3 end exponent text or end text V equals k T to the power of 2 divided by 3 end exponent... left parenthesis i right parenthesis text Differentiating equation (i), end text d V equals k 2 over 3 T to the power of negative 1 divided by 3 end exponent d T... left parenthesis i i right parenthesis text Work done end text d W equals P d V equals open parentheses fraction numerator R T over denominator V end fraction close parentheses times k 2 over 3 T to the power of negative 1 divided by 3 end exponent d T d W equals fraction numerator R T over denominator k T to the power of 2 divided by 3 end exponent end fraction times k times 2 over 3 T to the power of negative 1 divided by 3 end exponent d T equals 2 over 3 R d T... left parenthesis i i i right parenthesis W equals integral d W equals 2 over 3 integral subscript T subscript 1 end subscript superscript T subscript 2 end superscript d T equals 2 over 3 R open parentheses T subscript 2 minus T subscript 1 close parentheses equals 2 over 3 R straight capital delta T text Total work done, end text W equals 2 over 3 cross times R cross times 30 equals 20 R$

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